Godel’s Proof¨websupport1.citytech.cuny.edu/Faculty/tjohnstone/talks/godel.pdf · natural...
Transcript of Godel’s Proof¨websupport1.citytech.cuny.edu/Faculty/tjohnstone/talks/godel.pdf · natural...
Godel’s Proof
Victoria Gitman and Thomas Johnstone
New York City College of Technology, CUNY
[email protected]://websupport1.citytech.cuny.edu/faculty/vgitman
March 17, 2009
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Mathematical Logic
Mathematical Logic
Logic: Study of reasoning
Mathematical logic:
study of type of reasoning done by mathematicians
examines the methods used by mathematicians
Mathematics, as opposed to other sciences, uses proofs instead of observations.
impossible to prove all mathematical laws
certain first laws, axioms, are accepted without proof
the remaining laws, theorems, are to be proved from axioms
How do we accept certain axioms?
How do we choose reasonable axioms? Non-contradictory axioms? Powerfulaxioms?
What constitutes a proof from a given set of axioms? Which rules do we have dofollow at each step?
Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 2 / 21
Mathematical Logic
Gottlob Frege (1848-1925)
In his Begriffsschrift (1879), Frege
introduces symbolism for predicate logic
invents quantified variables: “for all”,and “there exists”
makes iterations of ∀ and ∃ understandable
“Every boy loves some girl” vs. “Some girl is loved by all boys”
invents axiomatic predicate logic
In his Grundgesetze der Arithmetik(1893, 1903), Frege attempts to axiomatize thetheory of sets.
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Mathematical Logic
Frege’s Set Building Axiom
“For any formal criterion, there exists a set whose members are those objects (andonly those objects) that satisfy the criterion.”
Frege’s axioms allows us to build various sets:
the set N of all natural numbers
the set R = {x : x is a real number}the set I = {x : x is an infinite set}the set of all sets, V = {x : x = x}
Some sets are members of themselves, while others are not!
Consider the set B of all objects that are not members of themselves, i.e.
B = {x : x /∈ x}
Question: What’s the problem with B?
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Mathematical Logic
Russell’s Paradox (1901)
Bertrand Russell (1872-1970) discovers that Frege’s axioms lead to a contradiction.
The key ideas:
The set B = {x : x /∈ x} cannot exist.
Self-reference: “x is not an element of itself”
Similar to the Liar Paradox (Epimenides, 400 BC):
This sentence is false!
Russell fixed Frege’s system using type theory.
This led to the ComprehensionAxiom and Zermelo-Fraenkel set theory.
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Formal Mathematics
Formal Mathematics
The 19th century work of Frege, Russell, Hilbert, Peano, Cantor, etc. leads todevelopment of formal systems:
A formal system consists of
A formal language
Axioms
Rulesof inference (how to conclude theorems from axioms)
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Formal Mathematics
Hilbert’s Program (1921)
David Hilbert (1862-1943) aimed to provide a secure foundation for mathematics.
Two Key Questions
Consistency: How do we know that contradictory consequences cannot be proved fromthe axioms?
Completeness: What if there are statements that cannot be decided by the axioms?
“No one shall expel us from the paradise that Cantor has created for us.”....Hilbert
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Formal Mathematics
Hilbert’s Program (continued...)
Translate all mathematics into a formallanguage and demonstrate “by finitary means” that
Peano Axioms (PA) for Number Theory,
Zermelo-Fraenkel (ZF) Axioms for Set Theory,
Euclidian Axioms for Geometry,
Principia Mathematica (PM) Axioms,
are consistent and complete!
What are Hilbert’s “ finitary means”?
Problem:
natural numbers, the universe of sets, the Euclidianspace are not finite!
Solution: Strip away the meaning of mathematical assertions!
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Formal Mathematics
Hilbert’s Program (continued...)
A mathematical assertion can be viewed in two fundamentally different ways:
as a sentence, namely a sequence of letters and symbols, or
as the meaning of the sentence.
Advantages of the syntactical view:
Mathematical concepts are very abstract
A sentence, studied as a syntactical object with no meaning, is very concrete.
Proofs are finite sequences of sentences that follow a few simple rules.
Provability can be studied without any understanding of the underlying subject.
“Mathematics is a game played according to certain simple rules with meaninglessmarks on paper.” ...Hilbert
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Formal Mathematics
Crash Course in Formal Mathematics
1) Logical symbols:
Equality: =
Boolean connectives: ∨, ∧, ¬,→Quantifiers: ∃, ∀
2) Functions, relations, and constants symbols: specific to subject
Number Theory: +, ·, <, 0, 1
Set Theory: ∈Group Theory: ◦, −1, e
3) Variables:x1, x2, x3, x4, . . . infinitely many!
4) Punctuation symbols:(, )
For notational convenience, we will write x , y , z instead of x1, x2, x3, respectively.
Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 10 / 21
Formal Mathematics
Writing Formal Mathematics: Formulas
Examples of formulas in Number Theory
x is even: even(x) := ∃y y + y = x
x divides y : x |y := ∃z z · x = y
x is prime: prime(x) := (∀y (y |x → (y = 1 ∨ y = x)) ∧ ¬x = 1)
3x = y : suggestions? (problem is that definition is recursive)
xy = z: (same problem)
3 is even: ∃x (x = (1 + 1) + 1 ∧ even(x))
There are infinitely many primes: ∀x∃y (y > x ∧ prime(y))
Every even number > 2 is the sum of two primes:∀x ((x > 1 + 1 ∧ even(x))→ ∃y∃z ((prime(y) ∧ prime(z)) ∧ x = y + x)))
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Formal Mathematics
Formulas (continued...)
So what is a formula?
Slogan: A formula, should if translated to English, correspond to a complete sentence.
Really, after stripping away any meaning:
A formula is a string of symbols built according to a finite set of simple rules.
This string is a formula: (why?)
∃z(z > 0 ∧ x + y = z)
This string is not a formula: (why not?)
∀x(y ∧ ∀z z > 0)
What are the rules?
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Formal Mathematics
Formula Witnessing Sequences
Recursive rules for building formulas:
Equality statements are formulas: x = y , x + y = z · zLess than statements are formulas: x + 1 < z
Boolean combinations of formulas are formulas:if ϕ and ψ are formulas, then so are(ϕ ∧ ψ), (ϕ ∨ ψ), ¬ϕ, (ϕ→ ψ).
A formula with a quantifier-variable pair attached in front is a formula:if i is any natural number and ϕ is a formula and , then so are ∃xi ϕ, ∀xi ϕ.
Nothing else is a formula
This recursive definition gives rise to formula witnessing sequences.
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Formal Mathematics
The Peano AxiomsAxiomatization of Number Theory proposed by Giuseppe Peano (1858-1932).
Peano AxiomsAddition and Multiplication
∀x∀y∀z (x + y) + z = x + (y + z) (associativity of addition)
∀x∀y x + y = y + x (commutativity of addition)
∀x∀y∀z (x · y) · z = x · (y · z) (associativity of multiplication)
∀x∀y x · y = y · x (commutativity of multiplication)
∀x∀y∀z x · (y + z) = x · y + x · z. (distributive law)
∀x (x + 0 = x ∧ x · 1 = x) (additive and multiplicative identity)
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Formal Mathematics
Peano Axioms (continued)Order
∀x∀y∀z ((x < y ∧ y < z)→ x < z) (the order is transitive)
∀x ¬x < x (the order is anti-reflexive)
∀x∀y ((x < y ∨ x = y) ∨ y < x) (any two elements are comparable)
∀x∀y∀z (x < y → x + z < y + z) (order respects addition)
∀x∀y∀z ((0 < z ∧ x < y)→ x · z < x · z) (order respects multiplication)
∀x∀y (x < y ↔ ∃z (z > 0 ∧ x + z = y))
∀x (x ≥ 0 ∧ (x > 0→ x ≥ 1)) (the order is discrete)
Induction SchemeFor every formula ϕ(x) we have
(ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1)))→ ∀xϕ(x)
Hilbert’s question: Are the Peano Axioms consistent? Are they complete?
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Formal Mathematics
The Group Theory Axioms: An Easy Example
Language: ◦, −1, e
Group Theory Axioms∀x∀y∀z x ◦ (y ◦ z) = (x ◦ y) ◦ z (associativity)
∀x (e ◦ x = x ∧ x ◦ e = x) (e is the identity)
∀x x ◦ x−1 = e (−1 is the inverse)
QuestionAre the group theory axioms consistent?
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Formal Mathematics
Group Theory Axioms: (continued...)
Hilbert would like:
Z4◦ e a b ce e a b ca a b c eb b c e ac c e a b
QuestionAre the group theory axioms complete?
Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 17 / 21
Formal Mathematics
Propositional LogicLanguage:
Variables: A, B, C, . . .Boolean connectives: ∧, ∨, ¬,→Punctuation symbols: (, )
Think of these variables as standing in for any sentence:
A =“Bush is a great public speaker”B =“It is going to rain tomorrow”C =“This talk is boring”
Rules for building formulas:A variable is a formula.Boolean combinations of formulas are formulas: (A→ B), ¬(A ∨ B)
The Rule of Inference is Modus Ponens: From A and A→ B, infer B.
Propositional Logic Axioms(A ∨ A)→ A
A→ (A→ B)
(A ∨ B)→ (B ∨ A)
(A→ B)→ ((C ∨ A)→ (C ∨ B))
Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 18 / 21
Formal Mathematics
Propositional Logic is Consistent
The following formulas are provable from the axioms:
A→ A
(A ∨ ¬A)
(A ∧ ¬A)→ B
The axioms are inconsistent if for some formula A, both A and ¬A are provable.
It suffices to find a single formula that is not provable!
Hilbert’s Strategy:
Find a structural, syntactical property that every derivable formula has.
Show that not every possible formula has that property.
What is the desired property?
The formula is a tautology, i.e. “true in all possible worlds”
Theorem (Hilbert, 1905)Propositional logic is consistent.
Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 19 / 21
Formal Mathematics
Presburger Arithmetic
Arithmetic without multiplication:
Presburger AxiomsAddition
∀x ¬0 = x + 1
∀x∀y (x + 1 = y + 1→ x = y)
∀x x + 0 = x
∀x∀y (x + y) + 1 = x + (y + 1)
Induction SchemeFor every formula ϕ(x) we have
(ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1)))→ ∀xϕ(x)
Mojzesz Presburger (1904-1943) showed in 1929 using finitary arguments thatPresburger Arithmetic is consistent and complete!
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Formal Mathematics
Godel ends Hilbert’s ProgramTheorem (Godel, 1931)The Peano Axioms are not complete. In fact, any “reasonable” collection of axioms forNumber Theory or Set Theory is necessarily incomplete.
Theorem (Godel, 1931)No proof of the consistency of the Peano Axioms can be given by “finitary means”.
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