The Formal Quantification of Infinities in the Modern Era

Cantor’s Theory of Sets and Transfinite Numbers

Dr. Jean Nicolas PestieauAssistant Professor of MathematicsState University of New York - Suffolk

5th Century B.C. The Greek philosopher Zeno of Elea proposes his paradoxes of motion, all rooted in some misconceptions of the infinite (but also in deeper questions relating to the nature of time and space...)

4th Century B.C. Aristotle’s work in metaphysics makes a distinction between the “potential” infinite and the “actual” infinite.

Early 17th Century Galileo shows the equinumerosity of the natural numbers with the perfect squares, leading to his celebrated paradox. His attempt to resolve this problem launches the first modern line of inquiry towards the infinite.

Late 17th Century Isaac Newton and Gottfried Leibniz independently develop the infinitesimal calculus, effectively paving the way for abstract analysis – a pillar of modern mathematics.

Historical Background (Pre-Cantor)

1870’s The Russian mathematician Georg Cantor proposes his ground-breaking theory of sets and an arithmetic for “transfinite” cardinal numbers. His work grounds the concept of infinity on a rigorous mathematical basis and equips mathematics with a firm logical foundation.

1900’s The British logician Bertrand Russell points to the simplest and most damaging paradox that emerges from Cantorian set theory.

1920’s The German mathematicians Ernst Zermelo and Abraham Fraenkel formulate an axiomatic theory of sets (known as ZFC when including the axiom of choice) and resolve all standing paradoxes.

1960’s - The American mathematician Paul Cohen builds on previous work by the Austrian logician Kurt Gödel to show that the continuum hypothesis is independent from ZFC.

Historical Background (Post-Cantor)

How can one explain the seeming absurdity of Zeno’s paradoxes?

How can one justify the infinitesimal processes at the heart of Newton’s and Leibniz’s calculus?

Why was Galileo correct in anticipating that “infinity should obey a different arithmetic than finite numbers?”

What is the significance of Cantor’s foundational work in set theory – and its implication that there exists “an infinity of infinities” – to modern mathematics and analytic philosophy?

Motivational Questions

In Zeno’s famous paradox, the philosopher argued that travel is an illusion.

Suppose you wish to reach a stationary point ahead. Then you would need to travel halfway there, and then one fourth of the way there, and then one eighth of the way there, and so on.

Because reaching your destination requires you to travel through an infinite number of points, Zeno concluded that motion is an impossibility – an uncanny conclusion!

The Dichotomy Paradox

A satisfactory answer to Zeno’s problem was reached in the early 19th

century when mathematicians – in particular Cauchy, Bolzano and

Weierstrass – established the modern definitions associated with infinitesimal processes.

These now constitute the basis of mathematical analysis, or abstract

analysis.

The most important of these definitions is that of a limit, which specifies

a mathematical object indirectly by means of a better and better approximation.

Resolving the Fallacy of the Dichotomy

Consider the following infinite sequence, where n is a positive integer,

which gives all the successive distances that you would need to travel

before arriving at your final destination. The sequence clearly approaches zero as n gets arbitrarily large, but does it ever reach zero?

“For all ∂ > 0, there exists a natural N such that, for all n ≥ N, < ∂.”

The affirmative answer implies that you eventually get to your destination and, therefore, disproves the paradox. This is precisely what analysis offers through a rigorous treatment of such an infinitesimal approximation.

In particular, here one would write , or equivalently

In the following discussion we shall extensively refer to infinite sets and their numerosity, or cardinality. Below are some basic definitions.

Def A set is a collection of well-defined, well-distinguished objects. These objects are then called the elements of the set.

Def For a given set S, the number of elements of S, denoted by |S|, is called the cardinal number, or cardinality, of S.

Def A set is called finite if its cardinality is a finite nonnegative integer.

Otherwise, the set is said to be infinite.

Preliminaries on Sets

Cantor’s work on sets is then a precise and systematic answer to the question “What are the cardinalities of infinite sets?”

Consider the set of natural numbers Ν = {1, 2, 3, 4, …} and the set of perfect squares (i.e. the squares of the naturals) S = {1, 4, 9, 16, 25, …}.

After careful thought, Galileo produced the following contradictory statements regarding these two sets:

1 – While some natural numbers are perfect squares, some are clearly not. Hence the set N must be more numerous than the set S, or |N|>|S|.

2 – Since for every perfect square there is exactly one natural that is its square

root, and for every natural there is exactly one perfect square, it follows that S and N are equinumerous, or |N|=|S|.

Galileo’s Paradox of Equinumerosity

To resolve the paradox, Galileo concluded that the concepts of “less,” “equal,” and “greater” were inapplicable to the cardinalities of infinite sets such as S and N, and could only be applied to finite sets.

In fact, Cantor would prove that, in general, this is not true. He showed that some infinite sets have a greater cardinality than others, thus implying the existence of different “sizes” for infinity.

Here, Galileo’s exact matching of the naturals with the perfect squares constitutes an early use of a one-to-one correspondence between sets – the conceptual basis for Cantor’s theory of sets.

What limited Galileo in his early study of number sets was the lack of a strict

methodology to compare the equinumerosity (or cardinality) of infinite sets.

Cantor was able to make his claims precise (that is, in step with the necessary

level of mathematical rigor) by establishing a formal framework for studying sets.

In particular, he defined equinumerosity in terms of one-to-one correspondences.

This resulted in a stronger definition of an infinite set.

The Framework for Cantor’s Theory of Sets

Def Two sets A and B are said to be in a one-to-one correspondenceif and only if there exists a bijection between the two sets. We then write A~B.

Def A set is said to be infinite if and only if it can be placed in a one-to-one correspondence with a proper subset of itself.

Cantor’s basic approach in developing his theory of sets was to systematically

relate the cardinality of various infinite number sets (in particular, the sets

of integer, rational and real numbers) to that of the naturals.

He started with the convention that |N|= (aleph-null) denotes the first transfinite cardinal, and then showed the hierarchy of

cardinal numbers:

This led to an extended transfinite arithmetic and also to the first conjecture

of the continuum hypothesis.

The Formal Quantification of Infinities: Cantor’s Cardinal Numbers

The Countability of I

Def A set A is said to be countable, or countably infinite if it can be placed in a one-to-one correspondence with N (i.e. iff |A|= ).

Otherwise the set is said to be uncountable.

Question: Is the set of all integers I = {…, -2, -1, 0, 1, 2, 3, …} countable or uncountable?

Answer: I~N, so |I|= and I is countable.

The Countability of Q

Consider the set of all rational numbers Q = {m/n| m and n are integers, n is nonzero}

Question: Is Q countable?

Answer: Yes! We can establisha one-to-one correspondence between the naturals and therationals and so Q~N. Once again, |Q|= .

Hilbert’s Paradox of the Grand Hotel In the early 1900’s, the German mathematician David Hilbert proposed a hypothetical scenario that keenly illustrates Cantor’s counter-intuitive results on infinite sets.

He called it the paradox of the Grand Hotel (or Hotel Infinity).

Consider a hotel with infinitely many rooms, all of which are occupied. He showed that it was possible then to accommodate a countably infinite number of passengers arriving at the hotel in a countably infinite number of buses!

His solution was, first, to empty the odd-numbered rooms by asking all guests in room N to move to room 2N. Then, he accommodated the passengers in the ith bus by assigning them to rooms pn, where n is 1, 2, 3, … and p is the (i+1)th prime number.

Are any infinite sets uncountable, or is the only size of infinity?

Hilbert’s result then leads to the following question:

The Uncountability of R: Cantor’s Diagonal Argument

One version of Cantor’s ingenious argument restricts R, the set of all real numbers, to U, the set of all real numbers between zero and one.

U = {x|x є R, 0 < x < 1}

Arguing by contradiction, suppose the list r1, r2, r3, …, rk, … enumeratesall the elements of U.

Then, using the decimal representation of real numbers, we write r1 = 0.d11d12d13…, r2 = 0.d21d22d23…, and, in general, rk = 0.dk1dk2dk3… for the kth real. Cantor was then able to construct a number not presentin this list by making the nth digit of this new number differ from dnn.

By extension, he concluded that |R|>|N| and, therefore, thatR is uncountable.

The Uncountability of R:Cantor’s Diagonal Argument

Transfinite Arithmetic

• If the cardinal numbers are finite, then the operations are those of ordinary arithmetic.

When Cantor proposed his cardinal numbers (informally, the sizes of allfinite or infinite sets), he also devised a new type of arithmetic for these generalized numbers.

• If the cardinal numbers are transfinite, then we have the following identities:

Hence we have, for example

(Cantor’s Power Set Theorem)

The Continuum Hypothesis

But is it the case that c = , the next cardinal number?

The affirmative answer to this question is what is known as the continuum hypothesis.

Cantor had stumbled upon this deep question, but failed to resolve it. It was only recently that Paul Cohen showed the independence of the continuum hypothesis from ZFC, the axiomatic theory of sets that would eventually replace Cantor’s theory of sets.

Cantor established that |N|=|I|=|Q|= and also that

c = |R|> , where c stands for continuum.

In other words, the continuum has a greater cardinality than the naturals, or any other countable set.

The Axiom of Choice (AC)

“The axiom of choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.” – Bertrand Russell

Informally, the axiom of choice says that we are allowed to make an arbitrary number of unspecified choices when forming a set.

It then justifies the action of forming from a set A the “set of all subsets of A” (Cantor’s power set theorem), or, say, allocating to all the members of a countable infinite family of countable sets a room in Hilbert’s Grand Hotel.

AC, in conjunction with the Zermelo-Fraenkel axioms of set theory, constitutes ZFC, the foundation of modern mathematics.

While it is widely regarded as such, many mathematicians and logicians feel uneasy about the nonconstructive nature of the formal proofs relying on AC.

The Banach-Tarski Paradox

The Banach-Tarski paradox relies on the axiom of choice to decompose a 3-dimensional unit ball into a finite number of disjoint pieces and then reassemble the pieces to form two unit balls.

This faux-paradox is counter-intuitive in the framework of Euclidian geometry, yet it is in fact consistent with the allowed construction of nonmeasurable sets in set-theoretical geometry.

This presentation was compiled from lecture notes I use for a course at Suffolk Community College titled A Survey of Mathematical Reasoning. A number of standard textbooks on mathematical concepts and ideas were referenced in writing those notes.

I also made multiple references to mathematical definitions, historical facts and illustrations that are readily accessible via internet on the online encyclopedia Wikipedia.

Finally, I used the book listed below as my primary reference.The Princeton Companion to Mathematics,

Timothy Gowers, Editor, June Barrow-Green and Imre Leader, Associate

Editors, Princeton University Press, 2008.

References

Thank you!