Principia MathematicaFrom Wikipedia, the free encyclopedia

Contents

1 Countable set 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Formal definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Empty product 92.1 Nullary arithmetic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Relevance of defining empty products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Nullary Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Nullary Cartesian product of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Nullary categorical product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 In computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Empty set 133.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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3.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Fuzzy set 194.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Fuzzy interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Fuzzy relation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Axiomatic definition of credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 Credibility inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.9 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.10 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Hereditarily finite set 265.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Ackermann’s bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Rado graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Infinite set 286.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Principia Mathematica 307.1 Scope of foundations laid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Theoretical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.2.1 Contemporary construction of a formal theory . . . . . . . . . . . . . . . . . . . . . . . . 31

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7.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2.3 Primitive ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.2.4 Primitive propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.3 Ramified types and the axiom of reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.4.1 An introduction to the notation of “Section A Mathematical Logic” (formulas ✸1–✸5.71) . 357.4.2 An introduction to the notation of “Section B Theory of Apparent Variables” (formulas ✸8–

✸14.34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.4.3 Introduction to the notation of the theory of classes and relations . . . . . . . . . . . . . . 39

7.5 Consistency and criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.5.1 Gödel 1930, 1931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.5.2 Wittgenstein 1919, 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.5.3 Gödel 1944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.6 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.6.1 Part I Mathematical logic. Volume I ✸1 to ✸43 . . . . . . . . . . . . . . . . . . . . . . . 427.6.2 Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97 . . . . . . . . . . . . . . 427.6.3 Part III Cardinal arithmetic. Volume II ✸100 to ✸126 . . . . . . . . . . . . . . . . . . . . 427.6.4 Part IV Relation-arithmetic. Volume II ✸150 to ✸186 . . . . . . . . . . . . . . . . . . . . 427.6.5 Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276 . . . . . . . . . . . 437.6.6 Part VI Quantity. Volume III ✸300 to ✸375 . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.7 Comparison with set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.8 Differences between editions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.10 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8 Recursive set 488.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9 Subset 509.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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10 Tarski–Grothendieck set theory 5410.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.2 Implementation in the Mizar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.3 Implementation in Metamath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11 Transitive set 5711.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.3 Transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.4 Transitive models of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

12 Uncountable set 5912.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

13 Universal set 6113.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13.1.1 Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.1.2 Cantor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

13.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 1

Countable set

“Countable” redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, theelements of a countable set can always be counted one at a time and, although the counting may never finish, everyelement of the set is associated with a natural number.Some authors use countable set to mean infinitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when finite sets are included and countably infinite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

1.1 Definition

A set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably infinite.In other words, a set is called “countably infinite” if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called “countablyinfinite,” and to not include finite sets.For alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function, seethe section Formal definition and properties below.

1.2 History

In the western world, different infinities were first classified by Georg Cantor around 1874.[5]

1.3 Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only effective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...”) is used, if the writer believes that the reader can easily guess what is missing; for example,

1

2 CHAPTER 1. COUNTABLE SET

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is finite.Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

YX123

x

246

2x

. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we first examine what it does not mean. For example, there are infinitely many oddintegers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... −2→−4,−1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

1.4 Formal definition and properties

By definition a set S is countable if there exists an injective function f : S → N from S to the natural numbers N ={0, 1, 2, 3, ...}.

1.4. FORMAL DEFINITION AND PROPERTIES 3

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all thesets containing two elements together; ...; finally, put together all infinite sets and consider them as having the samesize. This view is not tenable, however, under the natural definition of size.To elaborate this we need the concept of a bijection. Although a “bijection” seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory defines functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a↔ 1, b↔ 2, c↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines abijection.We now generalize this situation and define two sets to be of the same size if (and only if) there is a bijection betweenthem. For all finite sets this gives us the usual definition of “the same size”. What does it tell us about the size ofinfinite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for finite sets.Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for everypositive fraction, we can come up with a distinct number corresponding to it. This representation includes also thenatural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as manypositive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below(a more complex presentation is needed to deal with negative numbers).Theorem: The Cartesian product of finitely many countable sets is countable.This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedlymapping the first two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.Sometimes more than onemapping is useful. This is where youmap the set which you want to show countably infinite,onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers caneasily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q).What about infinite subsets of countably infinite sets? Do these have fewer elements than N?Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite setis countably infinite.For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

• 2 maps to 1

• 3 maps to 2

• 5 maps to 3

• 7 maps to 4

4 CHAPTER 1. COUNTABLE SET

1

2

3

01 2 31

2

3

4

5

6

7

8

9

11

12

13 18

17

24

0

0

The Cantor pairing function assigns one natural number to each pair of natural numbers

• 11 maps to 5

• 13 maps to 6

• 17 maps to 7

• 19 maps to 8

• 23 maps to 9

• ...

What about sets being “larger than” N? An obvious place to look would be Q, the set of all rational numbers, whichintuitively may seem much bigger than N. But looks can be deceiving, for we assert:Theorem: Q (the set of all rational numbers) is countable.Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto thesubset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} suchthat c = 0 if a/b ≥ 0 and c = 1 otherwise.

• 0 maps to (0,1,0)

1.4. FORMAL DEFINITION AND PROPERTIES 5

• 1 maps to (1,1,0)

• −1 maps to (1,1,1)

• 1/2 maps to (1,2,0)

• −1/2 maps to (1,2,1)

• 2 maps to (2,1,0)

• −2 maps to (2,1,1)

• 1/3 maps to (1,3,0)

• −1/3 maps to (1,3,1)

• 3 maps to (3,1,0)

• −3 maps to (3,1,1)

• 1/4 maps to (1,4,0)

• −1/4 maps to (1,4,1)

• 2/3 maps to (2,3,0)

• −2/3 maps to (2,3,1)

• 3/2 maps to (3,2,0)

• −3/2 maps to (3,2,1)

• 4 maps to (4,1,0)

• −4 maps to (4,1,1)

• ...

By a similar development, the set of algebraic numbers is countable, and so is the set of definable numbers.Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.For example, given countable sets a, b, c ...Using a variant of the triangular enumeration we saw above:

• a0 maps to 0

• a1 maps to 1

• b0 maps to 2

• a2 maps to 3

• b1 maps to 4

• c0 maps to 5

• a3 maps to 6

• b2 maps to 7

• c1 maps to 8

• d0 maps to 9

• a4 maps to 10

• ...

6 CHAPTER 1. COUNTABLE SET

Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore alsocountable by a previous theorem.Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...Theorem: The set of all finite-length sequences of natural numbers is countable.This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which isa countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which iscountable by the previous theorem.Theorem: The set of all finite subsets of the natural numbers is countable.If you have a finite subset, you can order the elements into a finite sequence. There are only countably many finitesequences, so also there are only countably many finite subsets.The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proofof this result can be found in Lang’s text.[2]

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S → N.

2. Either S is empty or there exists a surjective function g : N→ S.

3. Either S is finite or there exists a bijection h : N→ S.

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentationsee the sections above. Observe that N in the theorem can be replaced with any countably infinite set. In particularwe have the following Corollary.Corollary: Let S and T be sets.

1. If the function f : S → T is injective and T is countable then S is countable.

2. If the function g : S → T is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function h : T → N. Then if f : S → T is injectivethe composition h o f : S → N is injective, so S is countable.For (2) observe that if S is countable there is a surjective function h : N → S. Then if g : S → T is surjective thecomposition g o h : N→ T is surjective, so T is countable.Proposition: Any subset of a countable set is countable.Proof: The restriction of an injective function to a subset of its domain is still injective.Proposition: The Cartesian product of two countable sets A and B is countable.Proof: Note that N × N is countable as a consequence of the definition because the function f : N × N → N givenby f(m, n) = 2m3n is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian productof any two countable sets is countable. This follows because if A and B are countable there are surjections f : N→A and g : N→ B. So

f × g : N × N→ A × B

is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. This resultgeneralizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction onthe number of sets in the collection.Proposition: The integers Z are countable and the rational numbers Q are countable.Proof: The integers Z are countable because the function f : Z→ N given by f(n) = 2n if n is non-negative and f(n)= 3|n| if n is negative is an injective function. The rational numbers Q are countable because the function g : Z × N→ Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q.Proposition: If An is a countable set for each n in N then the union of all An is also countable.

1.5. MINIMAL MODEL OF SET THEORY IS COUNTABLE 7

Proof: This is a consequence of the fact that for each n there is a surjective function gn : N → An and hence thefunction

G : N× N→∪n∈N

An

given by G(n, m) = gn(m) is a surjection. Since N × N is countable, the Corollary implies that the union is countable.We are using the axiom of countable choice in this proof in order to pick for each n in N a surjection gn from thenon-empty collection of surjections from N to An.Cantor’s Theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is nosurjective function from A to P(A). A proof is given in the article Cantor’s Theorem. As an immediate consequenceof this and the Basic Theorem above we have:Proposition: The set P(N) is not countable; i.e. it is uncountable.For an elaboration of this result see Cantor’s diagonal argument.The set of real numbers is uncountable (see Cantor’s first uncountability proof), and so is the set of all infinitesequences of natural numbers. A topological proof for the uncountability of the real numbers is described at finiteintersection property.

1.5 Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standardmodel (see Constructible universe). The Löwenheim-Skolem theorem can be used to show that this minimal modelis countable. The fact that the notion of “uncountability” makes sense even in this model, and in particular that thismodel M contains elements which are

• subsets of M, hence countable,

• but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolem’s paradox.The minimal standard model includes all the algebraic numbers and all effectively computable transcendental num-bers, as well as many other kinds of numbers.

1.6 Total orders

Countable sets can be totally ordered in various ways, e.g.:

• Well orders (see also ordinal number):

• The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)• The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)

• Other (not well orders):

• The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)• The usual order of rational numbers (Cannot be explicitly written as a list!)

Note that in both examples of well orders here, any subset has a least element; and in both examples of non-wellorders, some subsets do not have a least element. This is the key definition that determines whether a total order isalso a well order.

8 CHAPTER 1. COUNTABLE SET

1.7 See also• Aleph number

• Counting

• Hilbert’s paradox of the Grand Hotel

• Uncountable set

1.8 Notes[1] For an example of this usage see (Rudin 1976, Chapter 2).

[2] See (Lang 1993, §2 of Chapter I).

[3] See (Apostol 1969, Chapter 13.19).

[4] Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no difference whether one considers 0 tobe a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic,which make 0 a natural number.

[5] Stillwell, John C. (2010), Roads to Infinity: TheMathematics of Truth and Proof, CRC Press, p. 10, ISBN 9781439865507,Cantor’s discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish betweencountable and uncountable infinities could not have been imagined.

1.9 References• Lang, Serge (1993), Real and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4

• Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X

• Apostol, Tom M. (June 1969),Multi-Variable Calculus and Linear Algebra with Applications, Calculus 2 (2nded.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5

1.10 External links• Weisstein, Eric W., “Countable Set”, MathWorld.

Chapter 2

Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by conventionequal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), justas the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3]

The term “empty product” is most often used in the above sense when discussing arithmetic operations. However,the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products incomputer programming; these are discussed below.

2.1 Nullary arithmetic product

2.1.1 Justification

Let a1, a2, a3,... be a sequence of numbers, and let

Pm =

m∏i=1

ai = a1 · · · am

be the product of the first m elements of the sequence. Then

Pm = am · Pm−1

for all m = 1,2,... provided that we use the following conventions: P1 = a1 and P0 = 1 . In other words, a “product”P1 with only one factor evaluates to that factor, while a “product” P0 with no factors at all evaluates to 1. Allowing a“product” with only one or zero factors reduces the number of cases to be considered in many mathematical formulas.Such “products” are natural starting points in induction proofs, as well as in algorithms. For these reasons, the “emptyproduct is one convention” is common practice in mathematics and computer programming.

2.1.2 Relevance of defining empty products

The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: whilethey seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentationof many subjects.For example, the empty products 0! = 1 and x0 = 1 shorten Taylor series notation (see zero to the power of zero for adiscussion when x=0). Likewise, if M is an n × n matrix then M0 is the n × n identity matrix, reflecting the fact thatapplying a linear map zero times has the same effect as applying the identity map.As another example, the fundamental theorem of arithmetic says that every positive integer can be written uniquelyas a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and itsproof!) become longer.[4][5]

9

10 CHAPTER 2. EMPTY PRODUCT

More examples of the use of the empty product in mathematics may be found in the binomial theorem (which assumesand implies that x0=1 for all x), Stirling number, König’s theorem, binomial type, binomial series, difference operatorand Pochhammer symbol.

2.1.3 Logarithms

Since logarithms turn products into sums, they should map an empty product to an empty sum. So if we define theempty product to be 1, then the empty sum should be ln(1) = 0 . Conversely, the exponential function turns sumsinto products, so if we define the empty sum to be 0, then the empty product should be e0 = 1 .

∏i

xi = e∑

i ln xi

2.2 Nullary Cartesian product

Consider the general definition of the Cartesian product:

∏i∈I

Xi = {g : I →∪i∈I

Xi | ∀i g(i) ∈ Xi}.

If I is empty, the only such g is the empty function f∅ , which is the unique subset of∅×∅ that is a function∅ → ∅, namely the empty subset ∅ (the only subset that ∅×∅ = ∅ has):

∏∅

= {f∅ : ∅ → ∅} = {∅}.

Thus, the cardinality of the Cartesian product of no sets is 1.Under the perhaps more familiar n-tuple interpretation,

∏∅

= {()},

that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality1.

2.2.1 Nullary Cartesian product of functions

The empty Cartesian product of functions is again the empty function.

2.3 Nullary categorical product

In any category, the product of an empty family is a terminal object of that category. This can be demonstrated byusing the limit definition of the product. An n-fold categorical product can be defined as the limit with respect to adiagram given by the discrete category with n objects. An empty product is then given by the limit with respect tothe empty category, which is the terminal object of the category if it exists. This definition specializes to give resultsas above. For example, in the category of sets the categorical product is the usual Cartesian product, and the terminalobject is a singleton set. In the category of groups the categorical product is the Cartesian product of groups, and theterminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product wemust take the decategorification of the empty product in the category of finite sets.Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may notexist in a given category; e.g. in the category of fields, neither exists.

2.4. IN LOGIC 11

2.4 In logic

Classical logic defines the operation of conjunction, which is generalized to universal quantification in and predicatecalculus, and is widely known as logical multiplication because we intuitively identify true with 1 and false with 0 andour conjunction behaves as ordinary multiplier. Multipliers can have arbitrary number of inputs. In case of 0 inputs,we have empty conjunction, which is identically equal to true.This is related to another concept in logic, vacuous truth, which tells us that empty set of objects can have anyproperty. It can be explained the way that the conjunction (as part of logic in general) deals with values less or equal1. This means that longer is the conjunction, the higher is probability to end up with 0. Conjunction merely checksthe propositions and returns 0 (or false) as soon as one of propositions evaluates to false. Reducing the number ofconjoined propositions increases the chance to pass the check and stay with 1. Particularly, if there are 0 tests ormembers to check, none can fail so, by default, we must always succeed regardless of which propositions or memberproperties had to be tested.

2.5 In computer programming

Many programming languages, such as Python, allow the direct expression of lists of numbers, and even functionsthat allow an arbitrary number of parameters. If such a language has a function that returns the product of all thenumbers in a list, it usually works like this:listprod( [2,3,5] ) --> 30 listprod( [2,3] ) --> 6 listprod( [2] ) --> 2 listprod( [] ) --> 1This convention helps avoid having to code special cases like “if length of list is 1” or “if length of list is zero” asspecial cases.Multiplication is an infix operator and therefore a binary operator, complicating the notation of an empty product.Some programming languages handle this by implementing variadic functions. For example, the fully parenthesizedprefix notation of Lisp languages gives rise to a natural notation for nullary functions:(* 2 2 2) ; evaluates to 8 (* 2 2) ; evaluates to 4 (* 2) ; evaluates to 2 (*) ; evaluates to 1

2.6 See also

• Iterated binary operation

• Empty sum

2.7 References

[1] Jaroslav Nešetřil, Jiří Matoušek (1998). Invitation to Discrete Mathematics. Oxford University Press. p. 12. ISBN 0-19-850207-9.

[2] A.E. Ingham and R C Vaughan (1990). The Distribution of Prime Numbers. Cambridge University Press. p. 1. ISBN0-521-39789-8.

[3] Page 9 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

[4] EdsgerWybe Dijkstra (1990-03-04). “How Computing Science created a newmathematical style”. EWD. Retrieved 2010-01-20. Hardy and Wright: “Every positive integer, except 1, is a product of primes”, Harold M. Stark: “If n is an integergreater than 1, then either n is prime or n is a finite product of primes.”. These examples —which I owe to A.J.M. vanGasteren— both reject the empty product, the last one also rejects the product with a single factor.

[5] Edsger Wybe Dijkstra (1986-11-14). “The nature of my research and why I do it”. EWD. Retrieved 2010-07-03. Butalso 0 is certainly finite and by defining the product of 0 factors —how else?— to be equal to 1 we can do away with theexception: “If n is a positive integer, then n is a finite product of primes.”

12 CHAPTER 2. EMPTY PRODUCT

2.8 External links• PlanetMath article on the empty product

Chapter 3

Empty set

"∅" redirects here. For similar symbols, see Ø (disambiguation).In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size orcardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists byincluding an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of setsare trivially true for the empty set.Null set was once a common synonym for “empty set”, but is now a technical term in measure theory. The empty setmay also be called the void set.

3.1 Notation

Common notations for the empty set include "{}", "∅", and " ∅ ". The latter two symbols were introduced by theBourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets(and not related in any way to the Greek letter Φ).[1]

The empty-set symbol ∅ is found at Unicode point U+2205.[2] In TeX, it is coded as \emptyset or \varnothing.

3.2 Properties

In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements;therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of “the emptyset” rather than “an empty set”.The mathematical symbols employed below are explained here.For any set A:

• The empty set is a subset of A:

∀A : ∅ ⊆ A

• The union of A with the empty set is A:

∀A : A ∪ ∅ = A

• The intersection of A with the empty set is the empty set:

∀A : A ∩ ∅ = ∅

• The Cartesian product of A and the empty set is the empty set:

∀A : A× ∅ = ∅

13

14 CHAPTER 3. EMPTY SET

The empty set is the set containing no elements.

The empty set has the following properties:

• Its only subset is the empty set itself:

∀A : A ⊆ ∅ ⇒ A = ∅

• The power set of the empty set is the set containing only the empty set:

2∅ = {∅}

3.2. PROPERTIES 15

A symbol for the empty set

• Its number of elements (that is, its cardinality) is zero:

card(∅) = 0

The connection between the empty set and zero goes further, however: in the standard set-theoretic definition ofnatural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.For any property:

• For every element of ∅ the property holds (vacuous truth);

• There is no element of ∅ for which the property holds.

Conversely, if for some property and some set V, the following two statements hold:

• For every element of V the property holds;

• There is no element of V for which the property holds,

V = ∅

16 CHAPTER 3. EMPTY SET

By the definition of subset, the empty set is a subset of any set A, as every element x of ∅ belongs to A. If it is nottrue that every element of ∅ is in A, there must be at least one element of ∅ that is not present in A. Since there areno elements of ∅ at all, there is no element of ∅ that is not in A. Hence every element of ∅ is in A, and ∅ is a subsetof A. Any statement that begins “for every element of ∅ " is not making any substantive claim; it is a vacuous truth.This is often paraphrased as “everything is true of the elements of the empty set.”

3.2.1 Operations on the empty set

Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sumof the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product).Ultimately, the results of these operations say more about the operation in question than about the empty set. Forinstance, zero is the identity element for addition, and one is the identity element for multiplication.A disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is adisarrangment of itself as no element can be found that retains its original position.

3.3 In other areas of mathematics

3.3.1 Extended real numbers

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member ofthat set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of thereal numbers, with its usual ordering, represented by the real number line, every real number is both an upper andlower bound for the empty set.[3] When considered as a subset of the extended reals formed by adding two “numbers”or “points” to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every otherextended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extendedreal number, then:

sup ∅ = min({−∞,+∞} ∪ R) = −∞,

and

inf ∅ = max({−∞,+∞} ∪ R) = +∞.

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound(inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinityis the identity element for the maximum and supremum operators, while positive infinity is the identity element forminimum and infimum.

3.3.2 Topology

Considered as a subset of the real number line (or more generally any topological space), the empty set is both closedand open; it is an example of a “clopen” set. All its boundary points (of which there are none) are in the empty set,and the set is therefore closed; while for every one of its points (of which there are again none), there is an openneighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the factthat every finite set is compact.The closure of the empty set is empty. This is known as “preservation of nullary unions.”

3.3.3 Category theory

If A is a set, then there exists precisely one function f from {} to A, the empty function. As a result, the empty set isthe unique initial object of the category of sets and functions.

3.4. QUESTIONED EXISTENCE 17

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the emptyset to be open. This empty topological space is the unique initial object in the category of topological spaces withcontinuous maps.The empty set is more ever a strict initial object: only the empty set has a function to the empty set.

3.4 Questioned existence

3.4.1 Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness followsfrom the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

• There is already an axiom implying the existence of at least one set. Given such an axiom together with theaxiom of separation, the existence of the empty set is easily proved.

• In the presence of urelements, it is easy to prove that at least one set exists, viz. the set of all urelements. Again,given the axiom of separation, the empty set is easily proved.

3.4.2 Philosophical issues

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity,whose meaning and usefulness are debated by philosophers and logicians.The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something.This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explainsthat the empty set is not nothing, but rather “the set of all triangles with four sides, the set of all numbers that arebigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king.”[4]

The popular syllogism

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sand-wich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darlingwrites that the contrast can be seen by rewriting the statements “Nothing is better than eternal happiness” and "[A]ham sandwich is better than nothing” in a mathematical tone. According to Darling, the former is equivalent to “Theset of all things that are better than eternal happiness is ∅ " and the latter to “The set {ham sandwich} is better thanthe set ∅ ". It is noted that the first compares elements of sets, while the second compares the sets themselves.[4]

Jonathan Lowe argues that while the empty set:

"...was undoubtedly an important landmark in the history of mathematics, … we should not assume thatits utility in calculation is dependent upon its actually denoting some object.”

it is also the case that:

“All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and(3) is unique amongst sets in having no members. However, there are very many things that 'have nomembers’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things haveno members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a setwhich has no members. We cannot conjure such an entity into existence by mere stipulation.”[5]

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtainedby plural quantification over individuals, without reifying sets as singular entities having other entities as members.[6]

18 CHAPTER 3. EMPTY SET

3.5 See also• Inhabited set

• Nothing

3.6 Notes[1] Earliest Uses of Symbols of Set Theory and Logic.

[2] Unicode Standard 5.2

[3] Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. Elementary Real Analysis, 2nd ed. Prentice Hall. P. 9.

[4] D. J. Darling (2004). The universal book of mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.

[5] E. J. Lowe (2005). Locke. Routledge. p. 87.

[6] • George Boolos, 1984, “To be is to be the value of a variable,” The Journal of Philosophy 91: 430–49. Reprinted inhis 1998 Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard Univ. Press: 54–72.

3.7 References• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

• Graham, Malcolm (1975), Modern Elementary Mathematics (HARDCOVER) (in English) (2nd ed.), NewYork: Harcourt Brace Jovanovich, ISBN 0155610392

3.8 External links• Weisstein, Eric W., “Empty Set”, MathWorld.

Chapter 4

Fuzzy set

In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced byLotfi A. Zadeh[1] and Dieter Klaua[2] in 1965 as an extension of the classical notion of set. At the same time, Salii(1965) defined a more general kind of structures called L-relations, which he studied in an abstract algebraic context.Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, et al., 2000), decision-making(Kuzmin, 1982) and clustering (Bezdek, 1978), are special cases of L-relations when L is the unit interval [0, 1].In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition— an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessmentof the membership of elements in a set; this is described with the aid of a membership function valued in the real unitinterval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases ofthe membership functions of fuzzy sets, if the latter only take values 0 or 1.[3] In fuzzy set theory, classical bivalentsets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which informationis incomplete or imprecise, such as bioinformatics.[4]

It has been suggested by Thayer Watkins that Zadeh’s ethnicity is an example of a fuzzy set because “His father wasTurkish-Iranian (Azerbaijani) and his mother was Russian. His father was a journalist working in Baku, Azerbaijanin the Soviet Union...Lotfi was born in Baku in 1921 and lived there until his family moved to Tehran in 1931.”[5]

4.1 Definition

A fuzzy set is a pair (U,m) where U is a set andm : U → [0, 1].

For each x ∈ U, the valuem(x) is called the grade of membership of x in (U,m). For a finite setU = {x1, . . . , xn},the fuzzy set (U,m) is often denoted by {m(x1)/x1, . . . ,m(xn)/xn}.Let x ∈ U. Then x is called not included in the fuzzy set (U,m) if m(x) = 0 , x is called fully included ifm(x) = 1 , and x is called a fuzzy member if 0 < m(x) < 1 .[6] The set {x ∈ U | m(x) > 0} is called thesupport of (U,m) and the set {x ∈ U | m(x) = 1} is called its kernel or core. The function m is called themembership function of the fuzzy set (U,m).

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a(fixed or variable) algebra or structureL of a given kind; usually it is required thatL be at least a poset or lattice. Theseare usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membershipfunctions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizationswere first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[7]

4.2 Fuzzy logic

Main article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations ( µ : Vo → W ) of propositional variables ( Vo ) into aset of membership degrees (W ) can be thought of as membership functions mapping predicates into fuzzy sets (or

19

20 CHAPTER 4. FUZZY SET

more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logiccan be extended to allow for fuzzy premises from which graded conclusions may be drawn.[8]

This extension is sometimes called “fuzzy logic in the narrow sense” as opposed to “fuzzy logic in the wider sense,”which originated in the engineering fields of automated control and knowledge engineering, and which encompassesmany topics involving fuzzy sets and “approximated reasoning.”[9]

Industrial applications of fuzzy sets in the context of “fuzzy logic in the wider sense” can be found at fuzzy logic.

4.3 Fuzzy number

Main article: Fuzzy number

A fuzzy number is a convex, normalized fuzzy set A ⊆ R whose membership function is at least segmentallycontinuous and has the functional value µA(x) = 1 at precisely one element.This can be likened to the funfair game “guess your weight,” where someone guesses the contestant’s weight, withcloser guesses being more correct, and where the guesser “wins” if he or she guesses near enough to the contestant’sweight, with the actual weight being completely correct (mapping to 1 by the membership function).

4.4 Fuzzy interval

A fuzzy interval is an uncertain set A ⊆ R with a mean interval whose elements possess the membership functionvalue µA(x) = 1 . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentallycontinuous.[10]

4.5 Fuzzy relation equation

The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation,and A · R stands for the composition of A with R.

4.6 Axiomatic definition of credibility[11] Let A be a non-empty set and P(A) be the power set of A . The set function Cr is known as credibility measureif it satisfies following condition

• Axiom 1: Cr{A} = 1

• Axiom 2: If B is subset of C, then, Cr{B} ≤ Cr{C}

• Axiom 3: Cr{B}+ Cr{Bc} = 1

• Axiom 4: Cr{∪Ai} = supi(Cr(Ai)) , for any event Ai with supi Cr{Ai} < 0.5

Cr{B} indicates how frequently event B will occur.

4.7 Credibility inversion theorem[12] Let A be a fuzzy variable with membership function u. Then for any set B of real numbers, we have

Cr{A ∈ B} =1

2

(supt∈B

u(t) + 1− supt∈Bc

u(t)

)

4.8. EXPECTED VALUE 21

4.8 Expected Value[13] Let A be a fuzzy variable. Then the expected value is

E[A] =

∫ ∞

0

Cr{A ≥ t} dt−∫ 0

−∞Cr{A ≤ t} dt.

4.9 Entropy[14] Let A be a fuzzy variable with a continuous membership function. Then its entropy is

H[A] =

∫ ∞

−∞S(Cr{A ≥ t}) dt.

Where

S(y) = −y lny − (1− y) ln(1− y)

4.10 Generalizations

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were intro-duced in 1965, a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity,and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory,while others try to mathematically model imprecision and uncertainty in a different way (Burgin and Chunihin, 1997;Kerre, 2001; Deschrijver and Kerre, 2003).The diversity of such constructions and corresponding theories includes:

• interval sets (Moore, 1966),

• L-fuzzy sets (Goguen, 1967),

• flou sets (Gentilhomme, 1968),

• Boolean-valued fuzzy sets (Brown, 1971),

• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),

• set-valued sets (Chapin, 1974; 1975),

• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),

• functions as generalizations of fuzzy sets and multisets (Lake, 1976),

• level fuzzy sets (Radecki, 1977)

• underdetermined sets (Narinyani, 1980),

• rough sets (Pawlak, 1982),

• intuitionistic fuzzy sets (Atanassov, 1983),

• fuzzy multisets (Yager, 1986),

• intuitionistic L-fuzzy sets (Atanassov, 1986),

• rough multisets (Grzymala-Busse, 1987),

22 CHAPTER 4. FUZZY SET

• fuzzy rough sets (Nakamura, 1988),

• real-valued fuzzy sets (Blizard, 1989),

• vague sets (Wen-Lung Gau and Buehrer, 1993),

• Q-sets (Gylys, 1994)

• shadowed sets (Pedrycz, 1998),

• α-level sets (Yao, 1997),

• genuine sets (Demirci, 1999),

• soft sets (Molodtsov, 1999),

• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)

• blurry sets (Smith, 2004)

• L-fuzzy rough sets (Radzikowska and Kerre, 2004),

• generalized rough fuzzy sets (Feng, 2010)

• rough intuitionistic fuzzy sets (Thomas and Nair, 2011),

• soft rough fuzzy sets (Meng, Zhang and Qin, 2011)

• soft fuzzy rough sets (Meng, Zhang and Qin, 2011)

• soft multisets (Alkhazaleh, Salleh and Hassan, 2011)

• fuzzy soft multisets (Alkhazaleh and Salleh, 2012)

4.11 See also• Alternative set theory

• Defuzzification

• Fuzzy concept

• Fuzzy mathematics

• Fuzzy measure theory

• Fuzzy set operations

• Fuzzy subalgebra

• Linear partial information

• Neuro-fuzzy

• Rough fuzzy hybridization

• Rough set

• Sørensen similarity index

• Type-2 Fuzzy Sets and Systems

• Uncertainty

• Interval finite element

• Multiset

4.12. REFERENCES 23

4.12 References[1] L. A. Zadeh (1965) “Fuzzy sets”. Information and Control 8 (3) 338–353.

[2] Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876.A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). “An early approach toward graded iden-tity and graded membership in set theory”. Fuzzy Sets and Systems 161 (18): 2369–2379. doi:10.1016/j.fss.2009.12.005.

[3] D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.

[4] Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, “FM-test: AFuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis”, BMC Bioinformatics, 7 (Suppl 4):S7. 2006.

[5] “Fuzzy Logic: The Logic of Fuzzy Sets”

[6] AAAI

[7] Goguen, Joseph A., 196, "L-fuzzy sets”. Journal of Mathematical Analysis and Applications 18: 145–174

[8] Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies PressLtd., ISBN 978-0-86380-262-1

[9] “The concept of a linguistic variable and its application to approximate reasoning,” Information Sciences 8: 199–249,301–357; 9: 43–80.

[10] “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems 1: 3–28

[11] Liu, Baoding. “Uncertain theory: an introduction to its axiomatic foundations.” Berlin: Springer-Verlag (2004).

[12] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[13] Liu, Baoding, and Yian-Kui Liu. “Expected value of fuzzy variable and fuzzy expected value models.” Fuzzy Systems,IEEE Transactions on 10.4 (2002): 445-450.

[14] Xuecheng, Liu. “Entropy, distancemeasure and similarity measure of fuzzy sets and their relations.” Fuzzy sets and systems52.3 (1992): 305-318.

4.13 Further reading

• Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID350600, 20 p.

• Alkhazaleh, S., Salleh, A.R. and Hassan, N. Soft Multisets Theory, Applied Mathematical Sciences, v. 5, No.72, 2011, pp. 3561–3573

• Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposited in Central Sci.-TechnicalLibrary of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)

• Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 87–96

• Bezdek, J.C. (1978) Fuzzy partitions and relations and axiomatic basis for clustering, Fuzzy Sets and Systems,v.1, pp. 111–127

• Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 77–97

• Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 32–39

• Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619–634

• Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255–267

• Chris Cornelis, Martine De Cock and Etienne E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads ofimperfect knowledge, Expert Systems, v. 20, issue 5, pp. 260–270, 2003

24 CHAPTER 4. FUZZY SET

• Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy settheory: construction, classification, application, International Journal of Approximate Reasoning, v. 35, pp.55–95

• Martine De Cock, Ulrich Bodenhofer, and Etienne E. Kerre, Modelling Linguistic Expressions Using FuzzyRelations, (2000) Proceedings 6th International Conference on Soft Computing. Iizuka 2000, Iizuka, Japan(1–4 October 2000) CDROM. p. 353-360

• Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377–384

• Deschrijver, G. and Kerre, E.E. On the relationship between some extensions of fuzzy set theory, Fuzzy Setsand Systems, v. 133, no. 2, pp. 227–235, 2003

• Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series7. Springer. ISBN 978-0-7923-7732-0.

• Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9,pp 899–911

• Gentilhomme, Y. (1968) Les ensembles flous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5,pp. 47–63

• Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145–174

• Gottwald, S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Basedand Axiomatic Approaches”. Studia Logica 82 (2): 211–244. doi:10.1007/s11225-006-7197-8.. Gottwald,S. (2006). “Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category TheoreticApproaches”. Studia Logica 84: 23–50. doi:10.1007/s11225-006-9001-1. preprint..

• Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math.Logik. Grundladen Math. 22, pp. 149–160.

• Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd InternationalSymposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325–332

• Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31.

• Ulrich Höhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measuretheory. The Handbooks of Fuzzy Sets Series 3. Springer. ISBN 978-0-7923-8388-8.

• Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115–132

• Kerre, E.E. A first view on the alternatives of fuzzy set theory, Computational Intelligence in Theory andPractice (B. Reusch, K-H . Temme, eds) Physica-Verlag, Heidelberg (ISBN 3-7908-1357-5), 2001, pp. 55–72

• George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN978-0-13-101171-7.

• Kuzmin,V.B. Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations, Nauka, Moscow, 1982(in Russian)

• Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323–326

• Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematicswith Applications’, v. 62, issue 12, 2011, pp. 4635–4645

• Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225–235,2001

• Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No.4/5, pp. 19–31

• Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966

• Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 1–8

4.14. EXTERNAL LINKS 25

• Narinyani, A.S. Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, ProjectVOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980

• Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, andCybernetics, Part B, 28, 103-109, 1998.

• Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics’, Volume 7, Issue 3-4, 1977

• Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Artificial Intelligence and SoftComputing - ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings;01/2004

• Salii, V.N. (1965) Binary L-relations, Izv. Vysh. Uchebn. Zaved., Matematika, v. 44, No.1, pp. 133–145 (inRussian)

• Sambuc, R. Fonctions φ-floues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. ThesisUniv. Marseille, France, 1975.

• Seising, Rudolf: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications—Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]:Springer 2007.

• Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165–235

• Thomas, K.V. and L. S. Nair, Rough intuitionistic fuzzy sets in a lattice, 'International Mathematical Forum',Vol. 6, 2011, no. 27, 1327 - 1335

• Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 23–37

• Yao, Y.Y., Combination of rough and fuzzy sets based on α-level sets, in: Rough Sets and Data Mining:Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp.301–321, 1997.

• Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1-4, 1998,pp. 227 – 242

• Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoning–I, Inform.Sci., v. 8, pp. 199–249

• Hans-Jürgen Zimmermann (2001). Fuzzy set theory—and its applications (4th ed.). Kluwer. ISBN 978-0-7923-7435-0.

• Gianpiero Cattaneo and Davide Ciucci, “Heyting Wajsberg Algebras as an Abstract Environment LinkingFuzzy andRough Sets” in J.J. Alpigini et al. (Eds.): RSCTC2002, LNAI 2475, pp. 77–84, 2002. doi:10.1007/3-540-45813-1_10

4.14 External links• Uncertainty model Fuzziness

• Fuzzy Systems Journal

• ScholarPedia

• The Algorithm of Fuzzy Analysis

• Fuzzy Image Processing

Chapter 5

Hereditarily finite set

“Nested set” redirects here. Nested set may also refer to the Nested set model in relational databases.In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily

V4 represented with circles in place of curly brackets

finite sets.

5.1 Formal definition

A recursive definition of well-founded hereditarily finite sets goes as follows:

Base case: The empty set is a hereditarily finite set.Recursion rule: If a1,...,ak are hereditarily finite, then so is {a1,...,ak}.

The set of all well-founded hereditarily finite sets is denoted Vω. If we denote P(S) for the power set of S, Vω canalso be constructed by first taking the empty set written V0, then V1 = P(V0), V2 = P(V1),..., Vk = P(Vk₋₁),... Then

∞∪k=0

Vk = Vω.

5.2 Discussion

The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consistingof the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinityis not a consequence of the other axioms of set theory.

26

5.3. ACKERMANN’S BIJECTION 27

Notice that there are countably many hereditarily finite sets, since Vn is finite for any finite n (its cardinality is n−12,see tetration), and the union of countably many finite sets is countable.Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. Vω is also symbolized by Hℵ0 ,meaning hereditarily of cardinality less than ℵ0 .

5.3 Ackermann’s bijection

Ackermann (1937) gave the following natural bijection f from the natural numbers to the hereditarily finite sets,known as the Ackermann coding. It is defined recursively by

f(2a + 2b + · · · ) = {f(a), f(b), . . .} if a, b, ... are distinct.

We have f(m)∈f(n) if and only if the mth binary digit of n (counting from the right starting at 0) is 1.

5.4 Rado graph

The graph whose vertices are the hereditarily finite sets, with an edge joining two vertices whenever one is containedin the other, is the Rado graph or random graph.

5.5 See also• Hereditarily countable set

5.6 References• Ackermann, Wilhelm (1937), “Die Widerspruchsfreiheit der allgemeinen Mengenlehre”, Mathematische An-nalen 114 (1): 305–315, doi:10.1007/BF01594179

Chapter 6

Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Someexamples are:

• the set of all integers, {..., −1, 0, 1, 2, ...}, is a countably infinite set; and

• the set of all real numbers is an uncountably infinite set.

6.1 Properties

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set thatis directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number.If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset.If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set isinfinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then atleast one of themmust be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian productof an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containingat least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.If an infinite set is a well-ordered set, then it must have a nonempty subset that has no greatest element.In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set, having a proper subsetequinumerous to itself.[1] If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

6.2 See also

• Aleph number

• Dedekind-infinite set

• Infinity

6.3 References[1] Boolos, George (1994), “The advantages of honest toil over theft”, Mathematics and mind (Amherst, MA, 1991), Logic

Comput. Philos., Oxford Univ. Press, New York, pp. 27–44, MR 1373892. See in particular pp. 32–33.

28

6.4. EXTERNAL LINKS 29

6.4 External links• Weisstein, Eric W., “Infinite Set”, MathWorld.

Chapter 7

Principia Mathematica

For Isaac Newton’s book containing basic laws of physics, see Philosophiæ Naturalis Principia Mathematica.The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North

54.43: “From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.” —Volume I, 1st edition,page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page86, accompanied by the comment, “The above proposition is occasionally useful.”)

Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second editionwith an important Introduction To the Second Edition, an Appendix A that replaced 9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic fromwhich all mathematical truths could in principle be proven. As such, this ambitious project is of great importance inthe history ofmathematics and philosophy,[1] being one of the foremost products of the belief that such an undertakingmay be achievable. However, in 1931, Gödel’s incompleteness theorem proved definitively that PM, and in fact anyother attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed toencapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematicswhich could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russelldiscovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out theunrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion ofa hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types.Contemporary mathematics, however, avoids paradoxes such as Russell’s in less unwieldy ways, such as the systemof Zermelo–Fraenkel set theory.PM is not to be confused with Russell’s 1903 Principles of Mathematics. PM states: “The present work was originallyintended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it becameincreasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental

30

7.1. SCOPE OF FOUNDATIONS LAID 31

questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe tobe satisfactory solutions.”The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentiethcentury.[2]

7.1 Scope of foundations laid

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems fromreal analysis were not included, but by the end of the third volume it was clear to experts that a large amount ofknown mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such adevelopment would be.A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustionupon completion of the third.

7.2 Theoretical basis

As noted in the criticism of the theory by Kurt Gödel (below), unlike a formalist theory, the “logicistic” theory ofPM has no “precise statement of the syntax of the formalism”. Another observation is that almost immediately in thetheory, interpretations (in the sense of model theory) are presented in terms of truth-values for the behaviour of thesymbols "⊢" (assertion of truth), "~" (logical not), and “V” (logical inclusive OR).Truth-values: PM embeds the notions of “truth” and “falsity” in the notion “primitive proposition”. A raw (pure)formalist theory would not provide the meaning of the symbols that form a “primitive proposition”—the symbolsthemselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behavebased on the grammar of the theory. Then later, by assignment of “values”, a model would specify an interpretation ofwhat the formulas are saying. Thus in the formal Kleene symbol set below, the “interpretation” of what the symbolscommonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)". But this isnot a pure Formalist theory.

7.2.1 Contemporary construction of a formal theory

The following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal systemwould be constructed as follows:

1. Symbols used: This set is the starting set, and other symbols can appear but only by definition from thesebeginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols "→"(implies, IF-THEN, "⊃"), "&" (and), “V” (or), "¬" (not), "∀" (for all), "∃" (there exists); predicate symbol "="(equals); function symbols "+" (arithmetic addition), "∙" (arithmetic multiplication), "'" (successor); individualsymbol “0” (zero); variables "a", "b", "c", etc.; and parentheses "(" and ")".[3]

2. Symbol strings: The theory will build “strings” of these symbols by concatenation (juxtaposition).[4]

3. Formation rules: The theory specifies the rules of syntax (rules of grammar) usually as a recursive definition thatstarts with “0” and specifies how to build acceptable strings or “well-formed formulas” (wffs).[5] This includesa rule for “substitution”.[6] of strings for the symbols called “variables” (as opposed to the other symbol-types).

4. Transformation rule(s): The axioms that specify the behaviours of the symbols and symbol sequences.

5. Rule of inference, detachment, modus ponens : The rule that allows the theory to “detach” a “conclusion” fromthe “premises” that led up to it, and thereafter to discard the “premises” (symbols to the left of the line │, orsymbols above the line if horizontal). If this were not the case, then substitution would result in longer andlonger strings that have to be carried forward. Indeed, after the application of modus ponens, nothing is leftbut the conclusion, the rest disappears forever.

Contemporary theories often specify as their first axiom the classical or modus ponens or “the rule ofdetachment":

32 CHAPTER 7. PRINCIPIA MATHEMATICA

A, A ⊃ B│ B

The symbol "│" is usually written as a horizontal line, here "⊃" means “implies”. The symbols A and Bare “stand-ins” for strings; this form of notation is called an “axiom schema” (i.e., there is a countablenumber of specific forms the notation could take). This can be read in a manner similar to IF-THEN butwith a difference: given symbol string IF A and A implies B THEN B (and retain only B for further use).But the symbols have no “interpretation” (e.g., no “truth table” or “truth values” or “truth functions”)and modus ponens proceeds mechanistically, by grammar alone.

7.2.2 Construction

The theory of PM has both significant similarities, and similar differences, to a contemporary formal theory. Kleenestates that “this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intendedto be believed, or at least to be accepted as plausible hypotheses concerning the world”.[7] Indeed, unlike a Formalisttheory that manipulates symbols according to rules of grammar, PM introduces the notion of “truth-values”, i.e., truthand falsity in the real-world sense, and the “assertion of truth” almost immediately as the fifth and sixth elements inthe structure of the theory (PM 1962:4–36):

• 1. Variables

• 2. Uses of various letters

• 3. The fundamental functions of propositions: “the Contradictory Function” symbolised by "~" and the “LogicalSum or Disjunctive Function” symbolised by "∨" being taken as primitive and logical implication defined (thefollowing example also used to illustrate 9. Definition below) as

p ⊃ q .=. ~ p ∨ q Df. (PM 1962:11)and logical product defined as

p . q .=. ~(~p ∨ ~q) Df. (PM 1962:12)

• 4. Equivalence: Logical equivalence, not arithmetic equivalence: "≡" given as a demonstration of how thesymbols are used, i.e., “Thus ' p ≡ q ' stands for '( p ⊃ q ) . ( q ⊃ p )'.” (PM 1962:7). Notice that to discuss anotation PM identifies a “meta"-notation with "[space] ... [space]":[8]

Logical equivalence appears again as a definition:

p ≡ q .=. ( p ⊃ q ) . ( q ⊃ p ) (PM 1962:12),

Notice the appearance of parentheses. This grammatical usage is not specified and appears sporadi-cally; parentheses do play an important role in symbol strings, however, e.g., the notation "(x)" for thecontemporary "∀x".

• 5. Truth-values: “The 'Truth-value' of a proposition is truth if it is true, and falsehood if it is false” (this phraseis due to Frege) (PM 1962:7).

• 6. Assertion-sign: "'⊦'. p may be read 'it is true that' ... thus '⊦: p .⊃. q ' means 'it is true that p implies q ',whereas '⊦. p .⊃⊦. q ' means ' p is true; therefore q is true'. The first of these does not necessarily involve thetruth either of p or of q, while the second involves the truth of both” (PM 1962:92).

• 7. Inference: PM 's version of modus ponens. "[If] '⊦. p ' and '⊦ (p ⊃ q)' have occurred, then '⊦ . q ' will occurif it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole recordis the occurrence of '⊦. q ' [in other words, the symbols on the left disappear or can be erased]" (PM 1962:9).

• 8. The Use of Dots

• 9. Definitions: These use the "=" sign with “Df” at the right end.

• 10. Summary of preceding statements: brief discussion of the primitive ideas "~ p" and "p ∨ q" and "⊦" prefixedto a proposition.

7.2. THEORETICAL BASIS 33

• 11. Primitive propositions: the axioms or postulates. This was significantly modified in the 2nd edition.

• 12. Propositional functions: The notion of “proposition” was significantly modified in the 2nd edition, includingthe introduction of “atomic” propositions linked by logical signs to form “molecular” propositions, and the useof substitution of molecular propositions into atomic or molecular propositions to create new expressions.

• 13. The range of values and total variation

• 14. Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned inthe 2nd edition. In particular, the distinction between the concepts defined in sections 15. Definition and thereal variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.

• 17. Formal implication and formal equivalence

• 18. Identity

• 19. Classes and relations

• 20. Various descriptive functions of relations

• 21. Plural descriptive functions

• 22. Unit classes

7.2.3 Primitive ideas

Cf. PM 1962:90–94, for the first edition:

• (1) Elementary propositions.

• (2) Elementary propositions of functions.

• (3) Assertion: introduces the notions of “truth” and “falsity”.

• (4) Assertion of a propositional function.

• (5) Negation: “If p is any proposition, the proposition “not-p", or "p is false,” will be represented by "~p" ".

• (6)Disjunction: “If p and q are any propositons, the proposition "p or q, i.e., “either p is true or q is true,” wherethe alternatives are to be not mutually exclusive, will be represented by "p ∨ q" ".

• (cf. section B)

7.2.4 Primitive propositions

The first edition (see discussion relative to the second edition, below) begins with a definition of the sign "⊃"1.01. p ⊃ q .=. ~ p ∨ q. Df.1.1. Anything implied by a true elementary proposition is true. Pp modus ponens

( 1.11 was abandoned in the second edition.)1.2. ⊦: p ∨ p .⊃. p. Pp principle of tautology1.3. ⊦: q .⊃. p ∨ q. Pp principle of addition1.4. ⊦: p ∨ q .⊃. q ∨ p. Pp principle of permutation1.5. ⊦: p ∨ ( q ∨ r ) .⊃. q ∨ ( p ∨ r ). Pp associative principle1.6. ⊦:. q ⊃ r .⊃: p ∨ q .⊃. p ∨ r. Pp principle of summation1.7. If p is an elementary proposition, ~p is an elementary proposition. Pp1.71. If p and q are elementary propositions, p ∨ q is an elementary proposition. Pp1.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨

ψp is an elementary proposition. Pp

34 CHAPTER 7. PRINCIPIA MATHEMATICA

Together with the “Introduction to the Second Edition”, the second edition’s Appendix A abandons the entire section9. This includes six primitive propositions 9 through 9.15 together with the Axioms of reducibility.

The revised theory is made difficult by the introduction of the Sheffer stroke ("|") to symbolise “incompatibility” (i.e.,if both elementary propositions p and q are true, their “stroke” p | q is false), the contemporary logical NAND (not-AND). In the revised theory, the Introduction presents the notion of “atomic proposition”, a “datum” that “belongsto the philosophical part of logic”. These have no parts that are propositions and do not contain the notions “all”or “some”. For example: “this is red”, or “this is earlier than that”. Such things can exist ad finitum, i.e., evenan “infinite eunumeration” of them to replace “generality” (i.e., the notion of “for all”).[9] PM then “advance[s] tomolecular propositions” that are all linked by “the stroke”. Definitions give equivalences for "~", "∨", "⊃", and ".".The new introduction defines “elementary propositions” as atomic and molecular positions together. It then replacesall the primitive propositions 1.2 to 1.72 with a single primitive proposition framed in terms of the stroke:

“If p, q, r are elementary propositions, given p and p|(q|r), we can infer r. This is a primitive proposition.”

The new introduction keeps the notation for “there exists” (now recast as “sometimes true”) and “for all” (recastas “always true”). Appendix A strengths the notion of “matrix” or “predicative function” (a “primitive idea”, PM1962:164) and presents four new Primitive propositions as 8.1– 8.13.88. Multiplicative axiom120. Axiom of infinity

7.3 Ramified types and the axiom of reducibility

In simple type theory objects are elements of various disjoint “types”. Types are implicitly built up as follows. Ifτ1,...,τm are types then there is a type (τ1,...,τm) that can be thought of as the class of propositional functions ofτ1,...,τm (which in set theory is essentially the set of subsets of τ1×...×τm). In particular there is a type () of propo-sitions, and there may be a type ι (iota) of “individuals” from which other types are built. Russell and Whitehead’snotation for building up types from other types is rather cumbersome, and the notation here is due to Church.In the ramified type theory of PM all objects are elements of various disjoint ramified types. Ramified types areimplicitly built up as follows. If τ1,...,τm,σ1,...,σn are ramified types then as in simple type theory there is a type(τ1,...,τm,σ1,...,σn) of “predicative” propositional functions of τ1,...,τm,σ1,...,σn. However there are also ramifiedtypes (τ1,...,τm|σ1,...,σn) that can be thought of as the classes of propositional functions of τ1,...τm obtained frompropositional functions of type (τ1,...,τm,σ1,...,σn) by quantifying over σ1,...,σn. When n=0 (so there are no σs) thesepropositional functions are called predicative functions or matrices. This is can be confusing because current math-ematical practice does not distinguish between predicative and non-predicative functions, and in any case PM neverdefines exactly what a “predicative function” actually is: this is taken as a primitive notion. Russell and Whiteheadfound it impossible to develop mathematics while maintaining the difference between predicative and non-predicativefunctions, so introduced the axiom of reducibility, saying that for every non-predicative function there is a predicativefunction taking the same values. In practice this axiom essentially means that the elements of type (τ1,...,τm|σ1,...,σn)can be identified with the elements of type (τ1,...,τm), which causes the hierarchy of ramified types to collapse downto simple type theory. (Strictly speaking this is not quite correct, because PM allows two propositional functions tobe different even it they take the same values on all arguments; this differs from current mathematical practice whereone normally identifies two such functions.)In Zermelo set theory one can model the ramified type theory of PM as follows. One picks a set ι to be the typeof individuals. For example, ι might be the set of natural numbers, or the set of atoms (in a set theory with atoms)or any other set one is interested in. Then if τ1,...,τm are types, the type (τ1,...,τm) is the power set of the productτ1×...×τm, which can also be thought of informally as the set of (propositional predicative) functions from this productto a 2-element set {true,false}. The ramified type (τ1,...,τm|σ1,...,σn) can be modeled as the product of the type(τ1,...,τm,σ1,...,σn) with the set of sequences of n quantifiers (∀ or ∃) indicating which quantifier should be appliedto each variable σi. (One can vary this slightly by allowing the σs to be quantified in any order, or allowing them tooccur before some of the τs, but this makes little difference except to the bookkeeping.)

7.4. NOTATION 35

7.4 Notation

Main article: List of notation used in Principia Mathematica

One author[1] observes that “The notation in that work has been superseded by the subsequent development of logicduring the 20th century, to the extent that the beginner has trouble reading PM at all"; while much of the symboliccontent can be converted to modern notation, the original notation itself is “a subject of scholarly dispute”, and somenotation “embod[y] substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism”.[10]

Kurt Gödel was harshly critical of the notation:

“It is to be regretted that this first comprehensive and thorough-going presentation of amathematical logicand the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations(contained in 1– 21 of Principia [i.e., sections 1– 5 (propositional logic), 8–14 (predicate logicwith identity/equality), 20(introduction to set theory), and 21 (introduction to relations theory)]) thatit represents in this respect a considerable step backwards as compared with Frege. What is missing,above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omittedeven in cases where they are necessary for the cogency of the proofs”.[11]

This is reflected in the example below of the symbols "p", "q", "r" and "⊃" that can be formed into the string "p ⊃ q ⊃r". PM requires a definition of what this symbol-string means in terms of other symbols; in contemporary treatmentsthe “formation rules” (syntactical rules leading to “well formed formulas”) would have prevented the formation ofthis string.Source of the notation: Chapter I “Preliminary Explanations of Ideas and Notations” begins with the source of theelementary parts of the notation (the symbols =⊃≡−ΛVε and the system of dots):

“The notation adopted in the present work is based upon that of Peano, and the following explanationsare to some extent modeled on those which he prefixes to his Formulario Mathematico [i.e., Peano 1889].His use of dots as brackets is adopted, and so are many of his symbols” (PM 1927:4).[12]

PM changed Peano’s Ɔ to ⊃, and also adopted a few of Peano’s later symbols, such as ℩ and ι, and Peano’s habit ofturning letters upside down.PM adopts the assertion sign "⊦" from Frege’s 1879 Begriffsschrift:[13]

"(I)t may be read 'it is true that'"[14]

Thus to assert a proposition p PM writes:

"⊦. p.” (PM 1927:92)

(Observe that, as in the original, the left dot is square and of greater size than the period on the right.)Most of the rest of the notation in PM was invented by Whitehead.

7.4.1 An introduction to the notation of “Section A Mathematical Logic” (formulas 1–5.71)

PM 's dots[15] are used in a manner similar to parentheses. Each dot (or multiple dot) represents either a left or rightparenthesis or the logical symbol ∧. More than one dot indicates the “depth” of the parentheses, for example, ".", ":"or ":.", "::". However the position of the matching right or left parenthesis is not indicated explicitly in the notationbut has to be deduced from some rules that are complicated, confusing and sometimes ambiguous. Moreover whenthe dots stand for a logical symbol ∧ its left and right operands have to be deduced using similar rules. First one hasto decide based on context whether the dots stand for a left or right parenthesis or a logical symbol. Then one has todecide how far the other corresponding parenthesis is: here one carries on until one meets either a larger number ofdots, or the same number of dots next that have equal or greater “force”, or the end of the line. Dots next to the signs

36 CHAPTER 7. PRINCIPIA MATHEMATICA

⊃, ≡,∨, =Df have greater force than dots next to (x), (∃x) and so on, which have greater force than dots indicating alogical product ∧.Example 1. The line

*3.12⊢ : ~p . v . ~q . v . p . q

corresponds to

(((~p) v (~q)) v (p ∧ q))

where the colon represents the outer (), the next two dots represent the parentheses around ~p and ~q, the third dotrepresents the parentheses around p ∧ q, and the fourth dot (rather confusingly) represents the logical symbol ∧ ratherthan a pair of parentheses.Example 2, with double, triple, and quadruple dots:

*9·521⊢ : : (∃x). φx . ⊃ . q : ⊃ : . (∃x). φx . v . r : ⊃ . q v r

stands for

((((∃x)(φx)) ⊃ (q)) ⊃ ((((∃x) (φx)) v (r)) ⊃ (q v r)))

Example 3, with a double dot indicating a logical symbol (from volume 1, page 10):

p⊃q:q⊃r.⊃.p⊃r

stands for

(p⊃q) ∧ ((q⊃r)⊃(p⊃r))

where the double dot represents the logical symbol ∧, and its right operand consists of everything after it because ithas priority over the single dots.Later in section 14, brackets "[ ]" appear, and in sections 20 and following, braces "{ }" appear. Whether thesesymbols have specific meanings or are just for visual clarification is unclear. Unfortunately the single dot (but also":", ":.", "::", etc.) is also used to symbolise “logical product” (contemporary logical AND often symbolised by "&"or "∧").Logical implication is represented by Peano’s "Ɔ" simplified to "⊃", logical negation is symbolised by an elongatedtilde, i.e., "~" (contemporary "~" or "¬"), the logical OR by “v”. The symbol "=" together with “Df” is used toindicate “is defined as”, whereas in sections 13 and following, "=" is defined as (mathematically) “identical with”,i.e., contemporary mathematical “equality” (cf. discussion in section 13). Logical equivalence is represented by "≡"(contemporary “if and only if”); “elementary” propositional functions are written in the customary way, e.g., "f(p)",but later the function sign appears directly before the variable without parenthesis e.g., "φx", "χx", etc.Example, PM introduces the definition of “logical product” as follows:

3.01. p . q .=. ~(~p v ~q) Df.

where "p . q" is the logical product of p and q.

3.02. p ⊃ q ⊃ r .=. p ⊃ q . q ⊃ r Df.

This definition serves merely to abbreviate proofs.

Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitivetranslation can be given. However, because of criticisms such as that of Kurt Gödel below, the best contemporarytreatments will be very precise with respect to the “formation rules” (the syntax) of the formulas.The first formula might be converted into modern symbolism as follows:[16]

7.4. NOTATION 37

(p & q) = (~(~p v ~q))

alternately

(p & q) = (¬(¬p v ¬q))

alternately

(p ∧ q) = (¬(¬p v ¬q))

etc.The second formula might be converted as follows:

(p→ q→ r) = (p→ q) & (q→ r)

But note that this is not (logically) equivalent to (p→ (q→ r)) nor to ((p→ q) → r), and these two are not logicallyequivalent either.

7.4.2 An introduction to the notation of “Section B Theory of Apparent Variables” (for-mulas 8– 14.34)

These sections concern what is now known as Predicate logic, and Predicate logic with identity (equality).

• NB: As a result of criticism and advances, the second edition of PM (1927) replaces 9 with anew 8 (Appendix A). This new section eliminates the first edition’s distinction between real andapparent variables, and it eliminates “the primitive idea 'assertion of a propositional function'.[17]To add to the complexity of the treatment, 8 introduces the notion of substituting a “matrix”, andthe Sheffer stroke:

• Matrix: In contemporary usage, PM 's matrix is (at least for propositionalfunctions), a truth table, i.e., all truth-values of a propositional or predicatefunction.

• Sheffer stroke: Is the contemporary logical NAND (NOT-AND), i.e., “in-compatibility”, meaning:“Given two propositions p and q, then ' p | q ' means “proposition p isincompatible with proposition q, i.e., if both propositions p and q evaluateas false, then p | q evaluates as true.” After section 8 the Sheffer strokesees no usage.

Section 10: The existential and universal “operators”: PM adds "(x)" to represent the contemporary symbol-ism “for all x " i.e., " ∀x", and it uses a backwards serifed E to represent “there exists an x", i.e., "(Ǝx)", i.e., thecontemporary "∃x”. The typical notation would be similar to the following:

"(x) . φx" means “for all values of variable x, function φ evaluates to true”"(Ǝx) . φx" means “for some value of variable x, function φ evaluates to true”

Sections 10, 11, 12: Properties of a variable extended to all individuals: section 10 introduces the notionof “a property” of a “variable”. PM gives the example: φ is a function that indicates “is a Greek”, and ψ indicates “isa man”, and χ indicates “is a mortal” these functions then apply to a variable x. PM can now write, and evaluate:

(x) . ψx

The notation above means “for all x, x is a man”. Given a collection of individuals, one can evaluate the above formulafor truth or falsity. For example, given the restricted collection of individuals { Socrates, Plato, Russell, Zeus } theabove evaluates to “true” if we allow for Zeus to be a man. But it fails for:

38 CHAPTER 7. PRINCIPIA MATHEMATICA

(x) . φx

because Russell is not Greek. And it fails for

(x) . χx

because Zeus is not a mortal.Equipped with this notation PM can create formulas to express the following: “If all Greeks are men and if all menare mortals then all Greeks are mortals”. (PM 1962:138)

(x) . φx ⊃ ψx :(x). ψx ⊃ χx :⊃: (x) . φx ⊃ χx

Another example: the formula:

10.01. (Ǝx). φx . = . ~(x) . ~φx Df.

means “The symbols representing the assertion 'There exists at least one x that satisfies function φ' is defined by thesymbols representing the assertion 'It’s not true that, given all values of x, there are no values of x satisfying φ'".The symbolisms ⊃x and "≡x" appear at 10.02 and 10.03. Both are abbreviations for universality (i.e., for all) thatbind the variable x to the logical operator. Contemporary notation would have simply used parentheses outside of theequality ("=") sign:

10.02 φx ⊃x ψx .=. (x). φx ⊃ ψx Df

Contemporary notation: ∀x(φ(x) → ψ(x)) (or a variant)

10.03 φx ≡x ψx .=. (x). φx ≡ ψx Df

Contemporary notation: ∀x(φ(x) ↔ ψ(x)) (or a variant)

PM attributes the first symbolism to Peano.Section 11 applies this symbolism to two variables. Thus the following notations: ⊃x, ⊃y, ⊃x, y could all appear ina single formula.Section 12 reintroduces the notion of “matrix” (contemporary truth table), the notion of logical types, and in par-ticular the notions of first-order and second-order functions and propositions.New symbolism "φ ! x" represents any value of a first-order function. If a circumflex "^" is placed over a variable,then this is an “individual” value of y, meaning that "ŷ" indicates “individuals” (e.g., a row in a truth table); thisdistinction is necessary because of the matrix/extensional nature of propositional functions.Now equipped with the matrix notion, PM can assert its controversial axiom of reducibility: a function of one or twovariables (two being sufficient for PM 's use) where all its values are given (i.e., in its matrix) is (logically) equivalent("≡") to some “predicative” function of the same variables. The one-variable definition is given below as an illustrationof the notation (PM 1962:166–167):12.1 ⊢: (Ǝ f): φx .≡x. f ! x Pp;

Pp is a “Primitive proposition” (“Propositions assumed without proof”) (PM 1962:12, i.e.,contemporary “axioms”), adding to the 7 defined in section 1 (starting with 1.1modus po-nens). These are to be distinguished from the “primitive ideas” that include the assertion sign"⊢", negation "~", logical OR “V”, the notions of “elementary proposition” and “elementarypropositional function"; these are as close as PM comes to rules of notational formation, i.e.,syntax.

This means: “We assert the truth of the following: There exists a function f with the property that: given all values ofx, their evaluations in function φ (i.e., resulting their matrix) is logically equivalent to some f evaluated at those samevalues of x. (and vice versa, hence logical equivalence)". In other words: given a matrix determined by property φ

7.4. NOTATION 39

applied to variable x, there exists a function f that, when applied to the x is logically equivalent to the matrix. Or:every matrix φx can be represented by a function f applied to x, and vice versa.13: The identity operator "=" : This is a definition that uses the sign in two different ways, as noted by the quote

from PM:

13.01. x = y .=: (φ): φ ! x . ⊃ . φ ! y Df

means:

“This definition states that x and y are to be called identical when every predicative function satisfied byx is also satisfied by y ... Note that the second sign of equality in the above definition is combined with“Df”, and thus is not really the same symbol as the sign of equality which is defined.”

The not-equals sign "≠" makes its appearance as a definition at 13.02.14: Descriptions:

“A description is a phrase of the form “the term y which satisfies φŷ, where φŷ is some function satisfiedby one and only one argument.”[18]

From this PM employs two new symbols, a forward “E” and an inverted iota "ɿ". Here is an example:

14.02. E ! ( ɿy) (φy) .=: ( Ǝb):φy . ≡y . y = b Df.

This has the meaning:

“The y satisfying φŷ exists,” which holds when, and only when φŷ is satisfied by one value of y and byno other value.” (PM 1967:173–174)

7.4.3 Introduction to the notation of the theory of classes and relations

The text leaps from section 14 directly to the foundational sections 20GENERALTHEORYOFCLASSES and21 GENERAL THEORY OF RELATIONS. “Relations” are what known in contemporary set theory as ordered

pairs. Sections 20 and 22 introduce many of the symbols still in contemporary usage. These include the symbols"ε", "⊂", "∩", "∪", "–", "Λ", and “V": "ε" signifies “is an element of” (PM 1962:188); "⊂" ( 22.01) signifies “iscontained in”, “is a subset of"; "∩" ( 22.02) signifies the intersection (logical product) of classes (sets); "∪" ( 22.03)signifies the union (logical sum) of classes (sets); "–" ( 22.03) signifies negation of a class (set); "Λ" signifies the nullclass; and “V” signifies the universal class or universe of discourse.Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.)(PM 1962:188):

x ε α

“The use of single letter in place of symbols such as ẑ(φz) or ẑ(φ ! z) is practically almostindispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' xε α' will mean ' x is a member of the class α'". (PM 1962:188)

α ∪ –α = V

The union of a set and its inverse is the universal (completed) set.[19]

α ∩ –α = Λ

The intersection of a set and its inverse is the null (empty) set.

40 CHAPTER 7. PRINCIPIA MATHEMATICA

When applied to relations in section 23CALCULUSOFRELATIONS, the symbols "⊂", "∩", "∪", and "–" acquirea dot: for example: "⊍", "∸".[20]

The notion, and notation, of “a class” (set): In the first edition PM asserts that no new primitive ideas are necessaryto define what is meant by “a class”, and only two new “primitive propositions” called the axioms of reducibility forclasses and relations respectively (PM 1962:25).[21] But before this notion can be defined, PM feels it necessary tocreate a peculiar notation "ẑ(φz)" that it calls a “fictitious object”. (PM 1962:188)

⊢: x ε ẑ(φz) .≡. (φx)

“i.e., ' x is a member of the class determined by (φẑ)' is [logically] equivalent to ' x satisfies(φẑ),' or to '(φx) is true.'". (PM 1962:25)

At least PM can tell the reader how these fictitious objects behave, because “A class is wholly determinate when itsmembership is known, that is, there cannot be two different classes having he same membership” (PM 1962:26).This is symbolised by the following equality (similar to 13.01 above:

ẑ(φz) = ẑ(ψz) . ≡ : (x): φx .≡. ψx

“This last is the distinguishing characteristic of classes, and justifies us in treating ẑ(ψz) asthe class determined by [the function] ψẑ.” (PM 1962:188)

Perhaps the above can be made clearer by the discussion of classes in Introduction to the 2nd Edition, which dis-poses of the Axiom of Reducibility and replaces it with the notion: “All functions of functions are extensional” (PM1962:xxxix), i.e.,

φx ≡x ψx .⊃. (x): ƒ(φẑ) ≡ ƒ(ψẑ) (PM 1962:xxxix)

This has the reasonable meaning that “IF for all values of x the truth-values of the functions φ and ψ of x are [logically]equivalent, THEN the function ƒ of a given φẑ and ƒ of ψẑ are [logically] equivalent.” PM asserts this is “obvious":

“This is obvious, since φ can only occur in ƒ(φẑ) by the substitution of values of φ for p, q, r, ... in a[logical-] function, and, if φx ≡ ψx, the substitution of φx for p in a [logical-] function gives the sametruth-value to the truth-function as the substitution of ψx. Consequently there is no longer any reason todistinguish between functions classes, for we have, in virtue of the above,φx ≡x ψx .⊃. (x). φẑ = . ψẑ".

Observe the change to the equality "=" sign on the right. PM goes on to state that will continue to hang onto thenotation "ẑ(φz)", but this is merely equivalent to φẑ, and this is a class. (all quotes: PM 1962:xxxix).

7.5 Consistency and criticisms

According to Carnap's “Logicist Foundations of Mathematics”, Russell wanted a theory that could plausibly be saidto derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition tothe basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namelythe axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms,Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be surethat the formal statements even properly express statements of real analysis, so that statements depending on it couldnot be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell’s ramification of the theory of typeswas unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.Beyond the status of the axioms as logical truths, one can ask the following questions about any system such as PM:

• whether a contradiction could be derived from the axioms (the question of inconsistency), and

• whether there exists a mathematical statement which could neither be proven nor disproven in the system (thequestion of completeness).

7.5. CONSISTENCY AND CRITICISMS 41

Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms ofset theory. (See Hilbert’s second problem.) Russell and Whitehead suspected that the system in PM is incomplete:for example, they pointed out that it does not seem powerful enough to show that the cardinal ℵω exists. Howeverone can ask if some recursively axiomatizable extension of it is complete and consistent.

7.5.1 Gödel 1930, 1931

In 1930, Gödel’s completeness theorem showed that first-order predicate logic itself was complete in a much weakersense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some modelof the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since agiven system of axioms (such as those of Principia Mathematica) may have many models, in some of which a givenstatement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.Gödel’s incompleteness theorems cast unexpected light on these two related questions.Gödel’s first incompleteness theorem showed that any recursive extension of Principia could not be both consistentand complete for arithmetic statements. (As mentioned above, Principia itself was already known to be incompletefor some non-arithmetic statements.) According to the theorem, within every sufficiently powerful recursive logicalsystem (such as Principia), there exists a statement G that essentially reads, “The statement G cannot be proved.”Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; andif G is not provable, then it is true, and the system is therefore incomplete.Gödel’s second incompleteness theorem (1931) shows that no formal system extending basic arithmetic can be usedto prove its own consistency. Thus, the statement “there are no contradictions in the Principia system” cannot beproven in the Principia system unless there are contradictions in the system (in which case it can be proven both trueand false).

7.5.2 Wittgenstein 1919, 1939

By the second edition of PM, Russell had removed his axiom of reducibility to a new axiom (although he does notstate it as such). Gödel 1944:126 describes it this way: “This change is connected with the new axiom that functionscan occur in propositions only “through their values”, i.e., extensionally . . . [this is] quite unobjectionable even fromthe constructive standpoint . . . provided that quantifiers are always restricted to definite orders”. This change froma quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i.e. functionsof functions: “We can decide that mathematics is to confine itself to functions of functions which obey the aboveassumption” (PM 2nd Edition p. 401, Appendix C).This new proposal resulted in a dire outcome. An “extensional stance” and restriction to a second-order predicatelogic means that a propositional function extended to all individuals such as “All 'x' are blue” now has to list all ofthe 'x' that satisfy (are true in) the proposition, listing them in a possibly infinite conjunction: e.g. x1 ∧ x2 ∧ . . .∧ xn ∧ . . .. Ironically, this change came about as the result of criticism from Wittgenstein in his 1919 TractatusLogico-Philosophicus. As described by Russell in the Preface to the 2nd edition of PM:

“There is another course, recommended by Wittgenstein† (†Tractatus Logico-Philosophicus, *5.54ff)for philosophical reasons. This is to assume that functions of propositions are always truth-functions,and that a function can only occur in a proposition through its values. . . . [Working through theconsequences] it appears that everything in Vol. I remains true . . . the theory of inductive cardinalsand ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largelycollapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. AlsoCantor’s proof that 2n > n breaks down unless n is finite.” (PM 2nd edition reprinted 1962:xiv, also cfnew Appendix C).

In other words, the fact that an infinite list cannot realistically be specified means that the concept of “number” in theinfinite sense (i.e. the continuum) cannot be described by the new theory proposed in PM Second Edition.Wittgenstein in his Lectures on the Foundations of Mathematics, Cambridge 1939 criticised Principia on variousgrounds, such as:

• It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practicessuch as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia,

42 CHAPTER 7. PRINCIPIA MATHEMATICA

this would be treated as evidence of an error in Principia (e.g., that Principia did not characterise numbers oraddition correctly), not as evidence of an error in everyday counting.

• The calculating methods in Principia can only be used in practice with very small numbers. To calculate usinglarge numbers (e.g., billions), the formulae would become too long, and some short-cut method would have tobe used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental andhence questionable methods such as induction). So again Principia depends on everyday techniques, not viceversa.

Wittgenstein did, however, concede that Principiamay nonetheless make some aspects of everyday arithmetic clearer.

7.5.3 Gödel 1944

In his 1944 Russell’s mathematical logic, Gödel offers a “critical but sympathetic discussion of the logicistic order ofideas":[22]

“It is to be regretted that this first comprehensive and thorough-going presentation of amathematical logicand the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations(contained in *1-*21 of Principia) that it represents in this respect a considerable step backwards ascompared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism.Syntactical considerations are omitted even in cases where they are necessary for the cogency of theproofs . . . The matter is especially doubtful for the rule of substitution and of replacing defined symbolsby their definiens . . . it is chiefly the rule of substitution which would have to be proved” (Gödel1944:124)[23]

7.6 Contents

7.6.1 Part I Mathematical logic. Volume I 1 to 43

This section describes the propositional and predicate calculus, and gives the basic properties of classes, relations,and types.

7.6.2 Part II Prolegomena to cardinal arithmetic. Volume I 50 to 97

This part covers various properties of relations, especially those needed for cardinal arithmetic.

7.6.3 Part III Cardinal arithmetic. Volume II 100 to 126

This covers the definition and basic properties of cardinals. A cardinal is defined to be an equivalence class ofsimilar classes (as opposed to ZFC, where a cardinal is a special sort of von Neumann ordinal). Each type has its owncollection of cardinals associated with it, and there is a considerable amount of bookkeeping necessary for comparingcardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare differentdefinitions of finite and infinite cardinals. ✸120.03 is the Axiom of infinity.

7.6.4 Part IV Relation-arithmetic. Volume II 150 to 186

A “relation-number” is an equivalence class of isomorphic relations. PM defines analogues of addition, multiplication,and exponentiation for arbitrary relations. The addition and multiplication is similar to the usual definition of additionand multiplication of ordinals in ZFC, though the definition of exponentiation of relations in PM is not equivalent tothe usual one used in ZFC.

7.7. COMPARISON WITH SET THEORY 43

7.6.5 Part V Series. Volume II 200 to 234 and volume III 250 to 276

This covers series, which is PM’s term for what is now called a totally ordered set. In particular it covers complete se-ries, continuous functions between series with the order topology (though of course they do not use this terminology),well-ordered series, and series without “gaps” (those with a member strictly between any two given members).

7.6.6 Part VI Quantity. Volume III 300 to 375

This section constructs the ring of integers, the fields of rational and real numbers, and “vector-families”, which arerelated to what are now called torsors over abelian groups.

7.7 Comparison with set theory

This section compares the system in PM with the usual mathematical foundations of ZFC. The system of PM isroughly comparable in strength with Zermelo set theory (or more precisely a version of it where the axiom of sepa-ration has all quantifiers bounded).

• The system of propositional logic and predicate calculus in PM is essentially the same as that used now, exceptthat the notation and terminology has changed.

• The most obvious difference between PM and set theory is that in PM all objects belong to one of a number ofdisjoint types. This means that everything gets duplicated for each (infinite) type: for example, each type hasits own ordinals, cardinals, real numbers, and so on. This results in a lot of bookkeeping to relate the varioustypes with each other.

• In ZFC functions are normally coded as sets of ordered pairs. In PM functions are treated rather differently.First of all, “function” means “propositional function”, something taking values true or false. Second, functionsare not determined by their values: it is possible to have several different functions all taking the same values(for example, one might regard 2x+2 and 2(x+1) as different functions on grounds that the computer programsfor evaluating them are different). The functions in ZFC given by sets of ordered pairs correspond to whatPM call “matrices”, and the more general functions in PM are coded by quantifying over some variables. Inparticular PM distinguishes between functions defined using quantification and functions not defined usingquantification, whereas ZFC does not make this distinction.

• PM has no analogue of the axiom of replacement.

• PM emphasizes relations as a fundamental concept, whereas in current mathematical practice it is functionsrather than relations that are treated as more fundamental; for example, category theory emphasizes morphismsor functions rather than relations. (However there is an analogue of categories called allegories that modelsrelations rather than functions, and is quite similar to the type system of PM.)

• In PM, cardinals are defined as classes of similar classes, whereas in ZFC cardinals are special ordinals. In PMthere is a different collection of cardinals for each type with some complicated machinery for moving cardinalsbetween types, whereas in ZFC there is only 1 sort of cardinal. Since PM does not have any equivalent of theaxiom of replacement, it is unable to prove the existence of cardinals greater than ℵω.

• In PM ordinals are treated as equivalence classes of well-ordered sets, and as with cardinals there is a differentcollection of ordinals for each type. In ZFC there is only one collection of ordinals, usually defined as vonNeumann ordinals. One strange quirk of PM is that they do not have an ordinal corresponding to 1, whichcauses numerous unnecessary complications in their theorems. The definition of ordinal exponentiation αβ inPM is not equivalent to the usual definition in ZFC and has some rather undesirable properties: for example, itis not continuous in β and is not well ordered (so is not even an ordinal).

• The constructions of the integers, rationals and real numbers in ZFC have been streamlined considerably overtime since the constructions in PM.

44 CHAPTER 7. PRINCIPIA MATHEMATICA

7.8 Differences between editions

Apart from corrections of misprints, the main text of PM is unchanged between the first and second editions. In thesecond edition volumes 2 and 3 are essentially unchanged apart from a change of page numbering, but volume 1 hasfive new additions:

• A 54 page introduction by Russell describing the changes they would have made had they had more time andenergy. The main change he suggests is the removal of the controversial axiom of reducibility, though headmits that he knows no satisfactory substitute for it. He also seems more favorable to the idea that a functionshould be determined by its values (as is standard in current mathematical practice).

• Appendix A, numbered as *8, 15 pages about the Sheffer stroke.

• Appendix B, numbered as *89, discussing induction without the axiom of reducibility

• Appendix C, 8 pages discussing propositional functions

• An 8 page list of definitions at the end, giving a much-needed index to the 500 or so notations used.

In 1962 CUP published a shortened paperback edition containing parts of the second edition of volume 1: the newintroduction, the main text up to *56, and appendices A and C.

7.9 See also• Axiomatic set theory

• Begriffsschrift

• Boolean algebra (logic)

• Information Processing Language (first computational demonstration of theorems in PM)

7.10 Footnotes[1] Irvine, Andrew D. (1 May 2003). “Principia Mathematica (Stanford Encyclopedia of Philosophy)". Metaphysics Research

Lab, CSLI, Stanford University. Retrieved 5 August 2009.

[2] “The Modern Library’s Top 100 Nonfiction Books of the Century”. The New York Times Company. 30 April 1999.Retrieved 5 August 2009.

[3] This set is taken from Kleene 1952:69 substituting → for ⊃.

[4] Kleene 1952:71, Enderton 2001:15

[5] Enderton 2001:16

[6] This is the word used by Kleene 1952:78

[7] Quote from Kleene 1952:45. See discussion LOGICISM at pages 43–46.

[8] In his section 8.5.4 Groping towards metalogic Grattain-Guiness 2000:454ff discusses the American logicians’ criticalreception of the second edition of PM. For instance Sheffer “puzzled that ' In order to give an account of logic, we mustpresuppose and employ logic ' " (p. 452). And Bernstein ended his 1926 review with the comment that “This distinctionbetween the propositional logic as a mathematical system and as a language must be made, if serious errors are to beavoided; this distinction the Principia does not make” (p.454).

[9] This idea is due to Wittgenstein’s Tractatus. See the discussion at PM 1962:xiv–xv)

[10] http://plato.stanford.edu/entries/pm-notation/

[11] Kurt Gödel 1944 “Russell’s mathematical logic” appearing at page 120 in Feferman et al. 1990 Kurt Gödel Collected WorksVolume II, Oxford University Press, NY, ISBN 978-0-19-514721-6(v.2.pbk.) .

7.11. REFERENCES 45

[12] For comparison, see the translated portion of Peano 1889 in van Heijenoort 1967:81ff.

[13] This work can be found at van Heijenoort 1967:1ff.

[14] And see footnote, both at PM 1927:92

[15] The original typography is a square of a heavier weight than the conventional period.

[16] The first example comes from plato.stanford.edu (loc.cit.).

[17] page xiii of 1927 appearing in the 1962 paperback edition to 56.

[18] The original typography employs an x with a circumflex rather than ŷ; this continues below

[19] See the ten postulates of Huntington, in particular postulates IIa and IIb at PM 1962:205 and discussion at page 206.

[20] The "⊂" sign has a dot inside it, and the intersection sign "∩" has a dot above it; these are not available in the Arial UnicodeMS font.

[21] Wiener 1914 “A simplification of the logic of relations” (van Hejenoort 1967:224ff) disposed of the second of these whenhe showed how to reduce the theory of relations to that of classes

[22] Kleene 1952:46.

[23] Gödel 1944 Russell’s mathematical logic in Kurt Gödel: Collected Works Volume II, Oxford University Press, New York,NY, ISBN 978-0-19-514721-6.

7.11 References

Primary:

• Whitehead, Alfred North; Russell, Bertrand (1910), Principia mathematica 1 (1 ed.), Cambridge: CambridgeUniversity Press, JFM 41.0083.02

• Whitehead, Alfred North; Russell, Bertrand (1912), Principia mathematica 2 (1 ed.), Cambridge: CambridgeUniversity Press, JFM 43.0093.03

• Whitehead, Alfred North; Russell, Bertrand (1913), Principia mathematica 3 (1 ed.), Cambridge: CambridgeUniversity Press, JFM 44.0068.01

• Whitehead, Alfred North; Russell, Bertrand (1925), Principia mathematica 1 (2 ed.), Cambridge: CambridgeUniversity Press, ISBN 978-0521067911, JFM 51.0046.06

• Whitehead, Alfred North; Russell, Bertrand (1927), Principia mathematica 2 (2 ed.), Cambridge: CambridgeUniversity Press, ISBN 978-0521067911, JFM 53.0038.02

• Whitehead, Alfred North; Russell, Bertrand (1927), Principia mathematica 3 (2 ed.), Cambridge: CambridgeUniversity Press, ISBN 978-0521067911, JFM 53.0038.02

• Whitehead, Alfred North; Russell, Bertrand (1997) [1962], Principia mathematica to *56, Cambridge Mathe-matical Library, Cambridge: CambridgeUniversity Press, ISBN0-521-62606-4,MR1700771, Zbl 0877.01042

The first edition was reprinted in 2009 by Merchant Books, ISBN 978-1-60386-182-3, ISBN 978-1-60386-183-0,ISBN 978-1-60386-184-7.Secondary:

• Stephen Kleene 1952 Introduction to Meta-Mathematics, 6th Reprint, North-Holland Publishing Company,Amsterdam NY, ISBN 0-7204-2103-9.

• Stephen Cole Kleene; Michael Beeson (March 2009). Introduction to Metamathematics (Paperback ed.).Ishi Press. ISBN 978-0-923891-57-2.

• Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870–1940, Princeton University Press,Princeton N.J., ISBN 0-691-05857-1 (alk. paper).

46 CHAPTER 7. PRINCIPIA MATHEMATICA

• LudwigWittgenstein 2009Major Works: Selected Philosophical Writings, HarperrCollins, NY, NY, ISBN 978-0-06-155024-9. In particular:

Tractatus Logico-Philosophicus (Vienna 1918, original publication in German).

• Jean van Heijenoort editor 1967 From Frege to Gödel: A Source book in Mathematical Logic, 1879–1931, 3rdprinting, Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 (pbk.)

• Michel Weber and Will Desmond (eds.). Handbook of Whiteheadian Process Thought, Frankfurt / Lancaster,Ontos Verlag, Process Thought X1 & X2, 2008.

7.12 External links• Stanford Encyclopedia of Philosophy:

• Principia Mathematica—by A. D. Irvine.• The Notation in Principia Mathematica—by Bernard Linsky.

• Proposition ✸54.43 in a more modern notation (Metamath)

7.12. EXTERNAL LINKS 47

The title page of the shortened version of the Principia Mathematica to *56

Chapter 8

Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithmwhich terminates after a finite amount of time and correctly decides whether or not a given number belongs to theset.A more general class of sets consists of the recursively enumerable sets, also called semidecidable sets. For thesesets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithmmay give no answer (but not the wrong answer) for numbers not in the set.A set which is not computable is called noncomputable or undecidable.

8.1 Formal definition

A subset S of the natural numbers is called recursive if there exists a total computable function f such that f(x) = 1 ifx ∈ S and f(x) = 0 if x ∉ S . In other words, the set S is recursive if and only if the indicator function 1S is computable.

8.2 Examples

• Every finite or cofinite subset of the natural numbers is computable. This includes these special cases:

• The empty set is computable.• The entire set of natural numbers is computable.• Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbersless than a given natural number is computable.

• The set of prime numbers is computable.

• A recursive language is a recursive subset of a formal language.

• The set of Gödel numbers of arithmetic proofs described in Kurt Gödel’s paper “On formally undecidablepropositions of Principia Mathematica and related systems I"; see Gödel’s incompleteness theorems.

8.3 Properties

If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then A ∩ B, A ∪ Band the image of A × B under the Cantor pairing function are recursive sets.A setA is a recursive set if and only ifA and the complement ofA are both recursively enumerable sets. The preimageof a recursive set under a total computable function is a recursive set. The image of a computable set under a totalcomputable bijection is computable.

48

8.4. REFERENCES 49

A set is recursive if and only if it is at level Δ01 of the arithmetical hierarchy.A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set.The image of a computable set under a nondecreasing total computable function is computable.

8.4 References• Cutland, N. Computability. Cambridge University Press, Cambridge-New York, 1980. ISBN 0-521-22384-9;ISBN 0-521-29465-7

• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1;ISBN 0-07-053522-1

• Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag,Berlin, 1987. ISBN 3-540-15299-7

8.5 External links• Sakharov, Alex, “Recursive Set”, MathWorld.

Chapter 9

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

50

9.1. DEFINITIONS 51

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

9.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

9.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

9.3 Examples

• The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = {1, 2, 3} is a subset of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

52 CHAPTER 9. SUBSET

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

9.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumeratingS = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from{0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

9.5 See also• Containment order

9.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

9.7. EXTERNAL LINKS 53

C B A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

9.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

Chapter 10

Tarski–Grothendieck set theory

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck)is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distin-guished from other axiomatic set theories by the inclusion of Tarski’s axiom which states that for each set there is aGrothendieck universe it belongs to (see below). Tarski’s axiom implies the existence of inaccessible cardinals, pro-viding a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supportscategory theory.The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.

10.1 Axioms

Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski’s axiom”.We will use the axioms, definitions, and notation of Mizar to describe it. Mizar’s basic objects and processes are fullyformal; they are described informally below. First, let us assume that:

• Given any set A , the singleton {A} exists.

• Given any two sets, their unordered and ordered pairs exist.

• Given any family of sets, its union exists.

TG includes the following axioms, which are conventional because they are also part of ZFC:

• Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC).

• Extensionality axiom: Two sets are identical if they have the same members.

• Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.

• Axiom schema of replacement: Let the domain of the function F be the set A . Then the range of F (thevalues of F (x) for all members x of A ) is also a set.

It is Tarski’s axiom that distinguishes TG from other axiomatic set theories. Tarski’s axiom also implies the axiomsof infinity, choice,[1][2] and power set.[3][4] It also implies the existence of inaccessible cardinals, thanks to which theontology of TG is much richer than that of conventional set theories such as ZFC.

• Tarski’s axiom (adapted from Tarski 1939[5]). For every set x , there exists a set y whose members include:

- x itself;- every subset of every member of y ;- the power set of every member of y ;

54

10.2. IMPLEMENTATION IN THE MIZAR SYSTEM 55

- every subset of y of cardinality less than that of y .More formally:

∀x∃y[x ∈ y ∧ ∀z ∈ y(P(z) ⊆ y ∧ P(z) ∈ y) ∧ ∀z ∈ P(y)(¬z ≈ y → z ∈ y)]

where " P(x) " denotes the power class of x and " ≈ " denotes equinumerosity. What Tarski’s axiom states (in thevernacular) for each set x there is a Grothendieck universe it belongs to.

10.2 Implementation in the Mizar system

The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the typesare assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty. The existence axioms, e.g. theexistence of the unordered pair, is also implemented indirectly by the definition of term constructors.The system includes equality, the membership predicate and the following standard definitions:

• Singleton: A set with one member;

• Unordered pair: A set with two distinct members. {a, b} = {b, a} ;

• Ordered pair: The set {{a, b}, {a}} = (a, b) = (b, a) ;

• Subset: A set all of whose members are members of another given set;

• The union of a family of sets Y : The set of all members of any member of Y .

10.3 Implementation in Metamath

The Metamath system supports arbitrary higher-order logics, but it is typically used with the “set.mm” definitions ofaxioms. The ax-groth axiom adds Tarski’s axiom, which in Metamath is defined as follows:⊢ ∃y(x ∈ y ∧ ∀z ∈ y (∀w(w ⊆ z → w ∈ y) ∧ ∃w ∈ y ∀v(v ⊆ z → v ∈ w)) ∧ ∀z(z ⊆ y → (z ≈ y ∨ z ∈ y)))

10.4 See also

• Axiom of limitation of size

10.5 Notes

[1] Tarski (1938)

[2] http://mmlquery.mizar.org/mml/current/wellord2.html#T26

[3] Robert Solovay, Re: AC and strongly inaccessible cardinals.

[4] Metamath grothpw.

[5] Tarski (1939)

56 CHAPTER 10. TARSKI–GROTHENDIECK SET THEORY

10.6 References• Andreas Blass, I.M. Dimitriou, and Benedikt Löwe (2007) "Inaccessible Cardinals without the Axiom ofChoice," Fundamenta Mathematicae 194: 179-89.

• Bourbaki, Nicolas (1972). “Univers”. In Michael Artin, Alexandre Grothendieck, Jean-Louis Verdier, eds.Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale desschémas – (SGA 4) – vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217.

• Patrick Suppes (1960) Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972.

• Tarski, Alfred (1938). "Über unerreichbare Kardinalzahlen” (PDF). Fundamenta Mathematicae 30: 68–89.

• Tarski, Alfred (1939). “On the well-ordered subsets of any set” (PDF). Fundamenta Mathematicae 32: 176–183.

10.7 External links• Trybulec, Andrzej, 1989, "Tarski–Grothendieck Set Theory", Journal of Formalized Mathematics.

• Metamath: "Proof Explorer Home Page." Scroll down to “Grothendieck’s Axiom.”

• PlanetMath: "Tarski’s Axiom"

Chapter 11

Transitive set

In set theory, a set A is transitive, if and only if

• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,• whenever x ∈ A, and x is not an urelement, then x is a subset of A.

Similarly, a class M is transitive if every element of M is a subset of M.

11.1 Examples

Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarilytransitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals).Any of the stages Vα and Lα leading to the construction of the von Neumann universe V and Gödel’s constructibleuniverse L are transitive sets. The universes L and V themselves are transitive classes.

11.2 Properties

A set X is transitive if and only if∪X ⊆ X , where

∪X is the union of all elements of X that are sets,

∪X = {y |

(∃x ∈ X)y ∈ x} . If X is transitive, then∪X is transitive. If X and Y are transitive, then X∪Y∪{X,Y} is transitive.

In general, if X is a class all of whose elements are transitive sets, then X ∪∪X is transitive.

A set X which does not contain urelements is transitive if and only if it is a subset of its own power set,X ⊂ P(X).The power set of a transitive set without urelements is transitive.

11.3 Transitive closure

The transitive closure of a set X is the smallest (with respect to inclusion) transitive set which contains X. Supposeone is given a set X, then the transitive closure of X is

∪{X,

∪X,

∪∪X,

∪∪∪X,

∪∪∪∪X, . . .}.

Note that this is the set of all of the objects related to X by the transitive closure of the membership relation.

11.4 Transitive models of set theory

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models.The reason is that properties defined by bounded formulas are absolute for transitive classes.

57

58 CHAPTER 11. TRANSITIVE SET

A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system.Transitivity is an important factor in determining the absoluteness of formulas.In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity.[1]

11.5 See also• End extension

• Transitive relation

• Supertransitive class

11.6 References[1] Goldblatt (1998) p.161

• Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society StudentTexts 39, Cambridge: Cambridge University Press, ISBN 0-521-59441-3, Zbl 0938.03067

• Goldblatt, Robert (1998), Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Textsin Mathematics 188, New York, NY: Springer-Verlag, ISBN 0-387-98464-X, Zbl 0911.03032

• Jech, Thomas (2008) [originally published in 1973], The Axiom of Choice, Dover Publications, ISBN 0-486-46624-8, Zbl 0259.02051

11.7 External links• Weisstein, Eric W., “Transitive”, MathWorld.

• Weisstein, Eric W., “Transitive Closure”, MathWorld.

• Weisstein, Eric W., “Transitive Reduction”, MathWorld.

Chapter 12

Uncountable set

“Uncountable” redirects here. For the linguistic concept, see Uncountable noun.

In mathematics, an uncountable set (or uncountably infinite set)[1] is an infinite set that contains too many elementsto be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinalnumber is larger than that of the set of all natural numbers.

12.1 Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of thefollowing conditions holds:

• There is no injective function from X to the set of natural numbers.

• X is nonempty and every ω-sequence of elements of X fails to include at least one element of X. That is, X isnonempty and there is no surjective function from the natural numbers to X.

• The cardinality of X is neither finite nor equal to ℵ0 (aleph-null, the cardinality of the natural numbers).

• The set X has cardinality strictly greater than ℵ0 .

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiomof choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

12.2 Properties

• If an uncountable set X is a subset of set Y, then Y is uncountable.

12.3 Examples

The best known example of an uncountable set is the set R of all real numbers; Cantor’s diagonal argument showsthat this set is uncountable. The diagonalization proof technique can also be used to show that several other sets areuncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of naturalnumbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or 2ℵ0 , or ℶ1

(beth-one).The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater thanzero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorffdimension strictly greater than zero must be uncountable.

59

60 CHAPTER 12. UNCOUNTABLE SET

Another example of an uncountable set is the set of all functions from R to R. This set is even “more uncountable”than R in the sense that the cardinality of this set is ℶ2 (beth-two), which is larger than ℶ1 .A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω1.The cardinality of Ω is denoted ℵ1 (aleph-one). It can be shown, using the axiom of choice, that ℵ1 is the smallestuncountable cardinal number. Thus either ℶ1 , the cardinality of the reals, is equal to ℵ1 or it is strictly larger. GeorgCantor was the first to propose the question of whether ℶ1 is equal to ℵ1 . In 1900, David Hilbert posed this questionas the first of his 23 problems. The statement that ℵ1 = ℶ1 is now called the continuum hypothesis and is known tobe independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

12.4 Without the axiom of choice

Without the axiom of choice, theremight exist cardinalities incomparable toℵ0 (namely, the cardinalities ofDedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth charac-terization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not wantto call them uncountable.If the axiom of choice holds, the following conditions on a cardinal κare equivalent:

• κ ≰ ℵ0;

• κ > ℵ0; and

• κ ≥ ℵ1 , where ℵ1 = |ω1| and ω1 is least initial ordinal greater than ω.

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriategeneralization of “uncountability” when the axiom fails. It may be best to avoid using the word in this case and specifywhich of these one means.

12.5 See also• Aleph number

• Beth number

• Injective function

• Natural number

12.6 References[1] Uncountably Infinite — from Wolfram MathWorld

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

12.7 External links• Proof that R is uncountable

Chapter 13

Universal set

For other uses, see Universal set (disambiguation).

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated,the conception of a universal set leads to a paradox (Russell’s paradox) and is consequently not allowed. However,some non-standard variants of set theory include a universal set.

13.1 Reasons for nonexistence

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, donot allow for the existence of a universal set. Its existence would cause paradoxes which would make the theoryinconsistent.

13.1.1 Russell’s paradox

Russell’s paradox prevents the existence of a universal set in Zermelo–Fraenkel set theory and other set theories thatinclude Zermelo's axiom of comprehension. This axiom states that, for any formula φ(x) and any set A, there existsanother set

{x ∈ A | φ(x)}

that contains exactly those elements x of A that satisfyφ . If a universal set V existed and the axiom of comprehensioncould be applied to it, then there would also exist another set {x ∈ V | x ∈ x} , the set of all sets that do not containthemselves. However, as Bertrand Russell observed, this set is paradoxical. If it contains itself, then it should notcontain itself, and vice versa. For this reason, it cannot exist.

13.1.2 Cantor’s theorem

A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power setis a set of sets, it would automatically be a subset of the set of all sets, provided that both exist. However, this conflictswith Cantor’s theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality thanthe set itself.

13.2 Theories of universality

The difficulties associated with a universal set can be avoided either by using a variant of set theory in which theaxiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.

61

62 CHAPTER 13. UNIVERSAL SET

13.2.1 Restricted comprehension

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V doesexist (and V ∈ V is true). In these theories, Zermelo’s axiom of comprehension does not hold in general, and theaxiom of comprehension of naive set theory is restricted in a different way. A set theory containing a universal set isnecessarily a non-well-founded set theory.The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. AlonzoChurch and Arnold Oberschelp also published work on such set theories. Church speculated that his theory mightbe extended in a manner consistent with Quine’s,[2] but this is not possible for Oberschelp’s, since in it the singletonfunction is provably a set,[3] which leads immediately to paradox in New Foundations.[4] The most recent advancesin this area have been made by Randall Holmes who published an online draft version of the book Elementary SetTheory with a Universal Set in 2012.[5]

13.2.2 Universal objects that are not sets

Main article: Universe (mathematics)

The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because mostversions of this theory do allow the use of quantifiers over all sets (see universal quantifier). One way of allowingan object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar largecollections as proper classes rather than as sets. One difference between a universal set and a universal class is thatthe universal class does not contain itself, because proper classes cannot be elements of other classes. Russell’sparadox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets aselements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, becauseit is not itself a set.

13.3 Notes[1] Forster 1995 p. 1.

[2] Church 1974 p. 308. See also Forster 1995 p. 136 or 2001 p. 17.

[3] Oberschelp 1973 p. 40.

[4] Holmes 1998 p. 110.

[5] http://math.boisestate.edu/~{}holmes/

13.4 References• Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedingsof Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297–308.

• T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides31). Oxford University Press. ISBN 0-19-851477-8.

• T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”• Bibliography: Set Theory with a Universal Set, originated by T. E. Forster and maintained by Randall Holmesat Boise State University.

• Randall Holmes (1998). Elementary Set theory with a Universal Set, volume 10 of the Cahiers du Centre deLogique, Academia, Louvain-la-Neuve (Belgium).

• Arnold Oberschelp (1973). “Set Theory over Classes,” Dissertationes Mathematicae 106.• WillardVanOrmanQuine (1937) “NewFoundations forMathematical Logic,”AmericanMathematicalMonthly44, pp. 70–80.

13.5. EXTERNAL LINKS 63

13.5 External links• Weisstein, Eric W., “Universal Set”, MathWorld.

64 CHAPTER 13. UNIVERSAL SET

13.6 Text and image sources, contributors, and licenses

13.6.1 Text• Countable set Source: https://en.wikipedia.org/wiki/Countable_set?oldid=665835456 Contributors: Damian Yerrick, AxelBoldt, Bryan

Derksen, Zundark, Xaonon, Danny, Oliverkroll, Kurt Jansson, Stevertigo, Patrick, Michael Hardy, Kku, Александър, Revolver, CharlesMatthews, Dysprosia, Hyacinth, Fibonacci, Head, Aleph4, Robbot, Romanm, MathMartin, Paul Murray, Ruakh, Tobias Bergemann,Giftlite, Everyking, Georgesawyer, Wyss, Simon Lacoste-Julien, Jorend, Hkpawn~enwiki, TheObtuseAngleOfDoom, Possession, Noisy,Rich Farmbrough, Pak21, Paul August, Gauge, Aranel, El C, PhilHibbs, Robotje, Jumbuck, Keenan Pepper, ABCD, Hu, Julioc, LOL,MattGiuca, Esben~enwiki, Graham87, Jetekus, Josh Parris, Salix alba, VKokielov, Kri, Chobot, Roboto de Ajvol, YurikBot, Taejo,Trovatore, Muu-karhu, Scs, Hirak 99, Lt-wiki-bot, Arthur Rubin, Reyk, Benandorsqueaks, Teply, Brentt, SmackBot, David Shear,Brick Thrower, Canthusus, Edonovan, Edgar181, Grokmoo, Xie Xiaolei, Silly rabbit, SEIBasaurus, Octahedron80, Javalenok, NYKevin,Matchups, SundarBot, Grover cleveland, Richard001, Bidabadi~enwiki, SashatoBot, Cronholm144, The Infidel, 16@r, Mets501, Newone,S0me l0ser, Martin Kozák, JRSpriggs, CRGreathouse, CBM, Mct mht, Gregbard, FilipeS, Thijs!bot, Colin Rowat, Magioladitis, Usien6,David Eppstein, MartinBot, Wdevauld, R'n'B, Ttwo, Qatter, Owlgorithm, Stokkink, Alyssa kat13, LokiClock, Rei-bot, Anonymous Dis-sident, Pieman93, Ilyaroz, Mike4ty4, SieBot, Caltas, Krishna.91, JackSchmidt, Unitvoice, Sunrise, Ken123BOT, Gigacephalus, Drag-onBot, Alexbot, Hans Adler, HumphreyW, Addbot, Topology Expert, Tomthecool, LaaknorBot, Super duper jimbo, LinkFA-Bot, Jarble,JakobVoss, Legobot, Luckas-bot, Yobot, AnomieBOT, Mihnea Maftei, Materialscientist, ArthurBot, Bdmy, Omnipaedista, Johnfranks,Worldrimroamer, Tkuvho, Phanxan, RedBot, Raiden09, EmausBot, QuantumOfHistory, Vishwaraj.anand00, Imcrazyaboutyou, AccessDenied, ClueBot NG, Wcherowi, Misshamid, Widr, Helpful Pixie Bot, Sinestar, Jim Sukwutput, Adityapanwarr, Dexbot, Sriharsh1234,Jochen Burghardt, Namespan, OliverBel, Matthew Kastor, Niceguy6, HKennethB and Anonymous: 127

• Empty product Source: https://en.wikipedia.org/wiki/Empty_product?oldid=655829085 Contributors: Damian Yerrick, Toby Bartels,Fubar Obfusco, Patrick, Michael Hardy, Komap, Paul A, Eric119, Ams80, Cyan, Revolver, Charles Matthews, WhisperToMe, McKay,Phil Boswell, Robbot, Fredrik, Altenmann, Henrygb, Wile E. Heresiarch, Giftlite, Paisa, ShaunMacPherson, Fropuff, Sundar, Cam-byses, Eequor, Fak119, Matt Crypto, CryptoDerk, Rlcantwell, Smyth, Paul August, Susvolans, Grick, Army1987, C S, La goutte depluie, Shreevatsa, Uncle G, Apokrif, MFH, Marudubshinki, Qwertyus, Jshadias, Chenxlee, Bubba73, Moskvax, Mathbot, Flashmor-bid, Trovatore, Nishantman, Ms2ger, WAS 4.250, Reyk, Bo Jacoby, SmackBot, InverseHypercube, Melchoir, Eskimbot, NoJoy, Oc-tahedron80, Javalenok, NYKevin, Daniel-Dane, Leland McInnes, Cybercobra, Daqu, MvH, EdC~enwiki, Happy-melon, Maxcantor,JRSpriggs, CBM, HenningThielemann, Cydebot, Headbomb, Dfrg.msc, RobHar, Ricardo sandoval, CommonsDelinker, Daniel5Ko, Os-sido, Steel1943, TXiKiBoT, Tom239, Anonymous Dissident, Dmcq, Thehotelambush, ClueBot, Watchduck, ChrisHodgesUK, Addbot,Ozob, Xario, ב ,.דניאל PV=nRT, Yobot, Citation bot, Charvest, D'ohBot, Citation bot 1, 777sms, Ebehn, Helpful Pixie Bot, Macofe andAnonymous: 43

• Empty set Source: https://en.wikipedia.org/wiki/Empty_set?oldid=666003197 Contributors: AxelBoldt, Lee Daniel Crocker, Uriyan,Bryan Derksen, Tarquin, Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Ryguasu, Hephaestos, Patrick,Michael Hardy,MartinHarper, TakuyaMurata, Eric119, Den fjättrade ankan~enwiki, Andres, Evercat, Renamed user 4, CharlesMatthews,Berteun, Dcoetzee, David Latapie, Dysprosia, Jitse Niesen, Krithin, Hyacinth, Spikey, Jeanmichel~enwiki, Flockmeal, Phil Boswell,Robbot, Sanders muc, Peak, Romanm, Gandalf61, Henrygb, Wikibot, Pengo, Tobias Bergemann, Adam78, Tosha, Giftlite, Dbenbenn,Vfp15, BenFrantzDale, Herbee, Fropuff, MichaelHaeckel, Macrakis, Python eggs, Rdsmith4, Mike Rosoft, Brianjd, Mormegil, Guan-abot, Paul August, Spearhead, EmilJ, BrokenSegue, Nortexoid, 3mta3, Obradovic Goran, Jonathunder, ABCD, Sligocki, Dzhim, Itsmine,HenryLi, Hq3473, Angr, Isnow, Qwertyus, MarSch, Salix alba, Bubba73, ChongDae, Salvatore Ingala, Chobot, YurikBot, RussBot,Rsrikanth05, Trovatore, Ms2ger, Saric, EtherealPurple, GrinBot~enwiki, TomMorris, SmackBot, InverseHypercube, Melchoir, FlashSh-eridan, Ohnoitsjamie, Joefaust, SMP, J. Spencer, Octahedron80, Iit bpd1962, Tamfang, Cybercobra, Dreadstar, RandomP, Jon Awbrey,Jóna Þórunn, Lambiam, Jim.belk, Vanished user v8n3489h3tkjnsdkq30u3f, Loadmaster, Hvn0413, Mets501, EdC~enwiki, Joseph Solisin Australia, Spindled, James pic, Amalas, Philiprbrenan, CBM, Gregbard, Cydebot, Pais, Julian Mendez, Malleus Fatuorum, Epbr123,Nick Number, Escarbot, Sluzzelin, .anacondabot, David Eppstein, Ttwo, Maurice Carbonaro, Ian.thomson, It Is Me Here, Daniel5Ko,NewEnglandYankee, DavidCBryant, VolkovBot, Zanardm, Rei-bot, Anonymous Dissident, Andy Dingley, SieBot, Niv.sarig, ToePeu.bot,Randomblue, Niceguyedc, Wounder, Nosolution182, Versus22, Palnot, AmeliaElizabeth, Feinoha, American Eagle, ThisIsMyWikipedi-aName, LaaknorBot, AnnaFrance, Numbo3-bot, Zorrobot, Legobot, Luckas-bot, Yobot, Ciphers, Xqbot, Nasnema, , GrouchoBot,LucienBOT, Pinethicket, Kiefer.Wolfowitz, Abductive, Jauhienij, FoxBot, Lotje, LilyKitty, Woodsy dong peep, EmausBot, Sharlack-Hames, Ystory, ClueBot NG, Cntras, Rezabot, Helpful Pixie Bot, Michael.croghan, Langing, Ugncreative Usergname, JYBot, Kephir,Phinumu, Noyster, GeoffreyT2000, Skw27 and Anonymous: 82

• Fuzzy set Source: https://en.wikipedia.org/wiki/Fuzzy_set?oldid=669262232 Contributors: Zundark, Taw, Toby Bartels, Boleslav Bob-cik, Michael Hardy, MartinHarper, Ixfd64, Tgeorgescu, Александър, AugPi, Palfrey, Evercat, Charles Matthews, Markhurd, Furrykef,Hyacinth, Grendelkhan, VeryVerily, Robbot, Jaredwf, Peak, Giftlite, Jcobb, Duncharris, JasonQuinn, Phe, Urhixidur, Elwikipedista~enwiki,El C, Kwamikagami, R. S. Shaw, Pinar, Kusma, Joriki, Smmurphy, Ryan Reich, Salix alba, Mathbot, Predictor, YurikBot, Wavelength,Michael Slone, SpuriousQ, Gaius Cornelius, Srinivasasha, Supten, Jurriaan, Ml720834~enwiki, SmackBot, Hydrogen Iodide, CommanderKeane bot, Dreadstar, Rijkbenik, Bjankuloski06en~enwiki, Valepert, Elharo, JRSpriggs, George100, Paulmlieberman, CRGreathouse,Ksoileau, Gregbard, VashiDonsk, NotQuiteEXPComplete, Helgus, Nick Number, Abdel Hameed Nawar, Михајло Анђелковић, MER-C, Ty580, Bouktin, Magioladitis, MartinBot, Maurice Carbonaro, Gerla, DoorsAjar, Krzysiulek~enwiki, BotKung, LBehounek, In-formationSpace, Kilmer-san, VanishedUserABC, Cesarpermanente, ClueBot, Lukipuk, QYV, Pgallert, Multipundit, Addbot, Wirelessfriend, Legobot, Yobot, AnomieBOT, DemocraticLuntz, Riyad parvez, Pownuk, J JMesserly, Charvest, T2gurut2, Kierkkadon, Tin-ton5, Carel.jonkhout, FoxBot, Mjs1991, DixonDBot, The tree stump, WikitanvirBot, Matsievsky, Tijfo098, ChuispastonBot, ClueBotNG, Dezireh batist, Frietjes, Helpful Pixie Bot, StarryGrandma, Zbhsueh, Dannyeuu, Jcallega, Mark viking, Faizan, DangerouslyPersua-siveWriter, Atharkharal, IITHemant, Reddraggone9, RudiSeising, JMP EAX, Ffswontforget3 and Anonymous: 92

• Hereditarily finite set Source: https://en.wikipedia.org/wiki/Hereditarily_finite_set?oldid=622366694Contributors: TheAnome,MichaelHardy, Rp, Angela, Revolver, Dysprosia, Greenrd, Onebyone, Army1987, Oleg Alexandrov, R.e.b., SmackBot, Mhss, Dreadstar, JR-Spriggs, CRGreathouse, CBM, Ariel., CommonsDelinker, Watchduck, Addbot, Reconsider the static, ClueBot NG, Pastisch and Anony-mous: 7

• Infinite set Source: https://en.wikipedia.org/wiki/Infinite_set?oldid=659075298Contributors: TheAnome, Toby Bartels, DennisDaniels,Charles Matthews, David Shay, Bkell, Tobias Bergemann, Giftlite, Paul August, Rgdboer, Benji22210, Zerofoks, Salix alba, FlaBot,

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VKokielov, Chobot, DVdm, 4C~enwiki, Grubber, Trovatore, Maksim-e~enwiki, Addshore, Bidabadi~enwiki, Vina-iwbot~enwiki, Lam-biam, StevenPatrickFlynn, Bjankuloski06en~enwiki, Fell Collar, JRSpriggs, CRGreathouse, CBM, Escarbot, JAnDbot, Olaf, David Epp-stein, Ttwo, Maurice Carbonaro, Doug, DFRussia, Cliff, JP.Martin-Flatin, Alexbot, Hans Adler, Hatsoff, Addbot, Favonian, Luckas-bot, TaBOT-zerem, AnomieBOT, JackieBot, Materialscientist, VladimirReshetnikov, Erik9bot, Nicolas Perrault III, BenzolBot, Tku-vho, Pinethicket, SkyMachine, TobeBot, Beyond My Ken, Wgunther, Tommy2010, ZéroBot, Donner60, ClueBot NG, Widr, Juro2351,Magic6ball, Sauood07, YFdyh-bot, Blackbombchu, K9re11, Centralpanic and Anonymous: 37

• PrincipiaMathematica Source: https://en.wikipedia.org/wiki/Principia_Mathematica?oldid=666278401Contributors: AxelBoldt, CYD,Vicki Rosenzweig, Deb, Miguel~enwiki, Edward, Wshun, Dominus, Chinju, Gaurav, Dcljr, Pde, Marco Krohn, Tim Retout, CharlesMatthews, Furrykef, VeryVerily, Bevo, Donreed, Goethean, Romanm, MathMartin, Rursus, LGagnon, Snobot, Ancheta Wis, Giftlite,Bfinn, Gro-Tsen, Gamaliel, Jorend, Varlaam, Neilc, APH,Maneesh, TylerMcHenry, Guanabot, Smyth, Paul August, Elwikipedista~enwiki,AJP, Crust, Mdd,Wricardoh~enwiki, Pgimeno~enwiki, Dirac1933, Kinema, OlegAlexandrov, Kzollman, Xaliqen, Tabletop, Cbustapeck,Teemu Leisti, Graham87, Qwertyus, Tim!, Danextpope, Salix alba, Nneonneo, R.e.b., Bubba73, SchuminWeb, Mathbot, Nihiltres, DavidH Braun (1964), Chobot, Jaraalbe, Bgwhite, Hairy Dude, Spl, Gaius Cornelius, KSchutte, Benja, Trovatore, Dilaudid~enwiki, GeoffCapp,Tomisti, Ilmari Karonen, SmackBot, Nick Dillinger, InverseHypercube, Agarvin, Logic2go, BiT, Jpvinall, Chris the speller, Clconway,Chlewbot, Jon Awbrey, Ohconfucius, Xdamr, Wvbailey, Filippowiki, Mets501, Novangelis, Dl2000, Vanished user, CBM, Drinibot,Thomasmeeks, Gregbard, Cydebot, BetacommandBot, Thijs!bot, Nick Number, RobotG, Magioladitis, David Eppstein, Exiledone,Msknathan, Robin S, Hysteron, Ttwo, Athaenara, Roelofs, Student7, Hugo999, VolkovBot, Nxavar, Caswick, Anonymous Dissident,BubbleDine, Bearian, Sapphic, SieBot, Pengyanan, Irober02, Sean.hoyland, ImageRemovalBot, Alpha Beta Epsilon, ChandlerMapBot,Auntof6, Alexbot, PixelBot, Hugo Herbelin, Palnot, Addbot, Cparkins111, Download, Abiyoyo, Lightbot, Drpickem, Luckas-bot, Yobot,Sprachpfleger, Ptbotgourou, Amirobot, AnomieBOT, Materialscientist, LilHelpa, MauritsBot, Xqbot, Moonpxi, , RibotBOT, Be-yond My Ken, EmausBot, John of Reading, Set theorist, Werieth, ZéroBot, Andemoreva, Landemor, Wcherowi, ODogerall, Lekrecteur-masque, Helpful Pixie Bot, BG19bot, Ceradon, Majingdong, The3me, Y256, Nhergert, Jodosma, OccultZone, Monkbot, Trackteur,Mrmodnar111, KasparBot and Anonymous: 87

• Recursive set Source: https://en.wikipedia.org/wiki/Recursive_set?oldid=662150948 Contributors: Michael Hardy, Docu, Ehn, Hy-acinth, Robbot, MathMartin, Ojigiri~enwiki, Saforrest, Tobias Bergemann, Filemon, Giftlite, Peruvianllama, Khalid hassani, Vivero~enwiki,Satyadev, Obradovic Goran, Arthena, Salix alba, NekoDaemon, BMF81, YurikBot, Hairy Dude, TheGrappler, Trovatore, R.e.s., ArthurRubin, Maksim-e~enwiki, Eskimbot, Mhss, Benkovsky, Adibob, Viebel, Feradz, Mets501, JRSpriggs, Ylloh, CBM, Gregbard, Cydebot,Thijs!bot, Ttwo, CBM2, Justin W Smith, DumZiBoT, BodhisattvaBot, Addbot, Yobot, Pcap, Almabot, VladimirReshetnikov, ZéroBot,AvicAWB, Fesharakif and Anonymous: 13

• Subset Source: https://en.wikipedia.org/wiki/Subset?oldid=670318766 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17 and Anonymous: 181

• Tarski–Grothendieck set theory Source: https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory?oldid=649863373Contributors: Dwheeler, Charles Matthews, R3m0t, Tobias Bergemann, Langec, RayBirks, EmilJ, Cherlin, Mdd, JosefUrban, Nmegill,Trovatore, SmackBot, JRSpriggs, CRGreathouse, CBM, Sam Staton, Jessealama, Wasell, Daniel5Ko, JohnBlackburne, JP.Martin-Flatin,DumZiBoT, Addbot, Yobot, EmausBot, UniversumExNihilo, Brad7777, K9re11 and Anonymous: 16

• Transitive set Source: https://en.wikipedia.org/wiki/Transitive_set?oldid=659275410 Contributors: Edward, Charles Matthews, JitseNiesen, Tobias Bergemann, Lethe, EmilJ, Oleg Alexandrov, Salix alba, Arthur Rubin, Mhss, Keithdunwoody, JRSpriggs, Vaughan Pratt,CBM, Gregbard, Roches, Ttwo, Franklin.vp, Addbot, Barak Sh, Luckas-bot, Xqbot, Erik9bot, Tkuvho, Wikielwikingo, EmausBot,ZéroBot, Deltahedron, Pastisch and Anonymous: 13

• Uncountable set Source: https://en.wikipedia.org/wiki/Uncountable_set?oldid=664962517Contributors: AxelBoldt, Tarquin, AstroNomer~enwiki,Taw, Toby Bartels, PierreAbbat, Patrick, Michael Hardy, Dominus, Kevin Baas, Revolver, Charles Matthews, Dysprosia, Hyacinth, Fi-bonacci, Aleph4, Robbot, Tobias Bergemann, Giftlite, Mshonle~enwiki, Fropuff, Noisy, Crunchy Frog, Func, Keenan Pepper, OlegAlexandrov, Graham87, Island, Salix alba, FlaBot, Margosbot~enwiki, YurikBot, Piet Delport, Gaius Cornelius, Trovatore, Scs, Bota47,Arthur Rubin, Naught101, SmackBot, Bh3u4m, Bananabruno, SundarBot, Dreadstar, Germandemat, Loadmaster, Mets501, Limaner,Stephen B Streater, JRSpriggs, CRGreathouse, CBM, Gregbard, Thijs!bot, Dugwiki, Salgueiro~enwiki, JAnDbot, .anacondabot, Ttwo,Qatter, KarenJo90, SieBot, Phe-bot, ClueBot, Canopus1, DumZiBoT,Addbot, Yobot, Omnipaedista, BenzolBot, FoxBot, Vishwaraj.anand00,Mark viking, ILLUSION-ZONE and Anonymous: 33

• Universal set Source: https://en.wikipedia.org/wiki/Universal_set?oldid=667869756 Contributors: Awaterl, Patrick, Charles Matthews,Dysprosia, Hyacinth, Paul August, Jumbuck, Gary, Salix alba, Chobot, Hairy Dude, SmackBot, Incnis Mrsi, FlashSheridan, Gilliam,Lambiam, AndriusKulikauskas, Newone, CBM, User6985, Cydebot, LookingGlass, David Eppstein, Ttwo, VolkovBot, Anonymous Dis-sident, SieBot, ToePeu.bot, Oxymoron83, Cliff, Addbot, Neodop, Download, Dimitris, Yobot, Shlakoblock, Citation bot, Xqbot, Amaury,FrescoBot, Aikidesigns, Petrb, Wcherowi, Jochen Burghardt, Vivianthayil, Smortypi, Blackbombchu, TerryAlex and Anonymous: 24

66 CHAPTER 13. UNIVERSAL SET

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