Infinity in the palm of your hand

Hannes Leitgeb and Stephan Hartmann

Personal notes from lectures of Week 1: Infinity
(These are personal summaries and paraphrasing of some of the materials in the lectures that I felt to be important, with possible additions of personal impressions. They are not meant to be comprehensive records nor intended to be reproductions of copyright materials.)

Let us see how set theory in mathematics can help us understand infinity. The Principle of Extensionality states, in essence, that two sets are identical if and only if they have the same members.

There are two different ways of comparing the size of sets. One can, for instance, say that if X is a proper subset of Y (all the members of X are members of Y, but not vice versa), then X is smaller than Y. Another way of doing so is to pair off members of the sets with each other. Set X {A,B} can be paired with Y {1,2} but not with Z {1,2,3} because Z has a member that has no corresponding pair. So X and Y are of equal size, but both are smaller than Z. Normally these two methods lead to the same result, but in case of infinite sets, this can lead to a paradox, first realized by Galileo.

P1. If X is a proper subset of Y, then X is smaller than Y.
P2. If there is a pairing off between members of X and Y, then they are of equal size.
P3. The set of even natural numbers {2, 4, 6…} is a proper subset of the set of positive natural numbers {1,2,3,4,5,6…}.
P4. There is a pairing off between the even natural numbers and the positive natural numbers. (Each even natural number can be paired with its half. 2-1, 4-2, 6-3,…)

From P1 and P3 we conclude that the set of even natural numbers is smaller than the set of positive natural numbers, while from P2 and P4 we conclude that the set of even natural numbers is equal in size to the set of positive natural numbers. This is a logical contradiction.

Galileo concluded from this that infinite sets cannot be compared in terms of size. That is, P1 and P2 do not apply to infinite sets. Another response is to distinguish between two senses of ‘equal size’. These two senses are different and distinct, such that in one sense the set of even natural number is smaller in size, while in another sense it is equal in size to the set of positive natural numbers. In this case we end up with two ways of extending the concept of size from finite to the infinite.

In turns out, however, that the way of pairing off members of sets offers us a precise, consistent and systematic way of building a notion of what it means for sets to be of equal size. The first sense only tells us about inequality, but not really about equality. The pairing off method can also be used to distinguish inequality. For example, X has less members than Y, if members of X can only be paired off with a proper subset of Y, and not with all members of Y. Therefore, this way of comparing set is the one we ought to utilize.

Using this way of comparing sets, we can even define what an infinite set is:
A set is infinite, if and only if all the members of the set can be paired off with at least one of its proper subsets.
(For example, the set of positive natural numbers can be paired off with the set of positive even natural numbers, which is its proper subset)

“Infinity amounts to a kind of self-similarity property. Much as with fractals in geometry. An infinite set contains something which, in terms of size, looks like itself.” (Lecturer’s words.)

The definition of infinite set was by Richard Dedekind, and Georg Cantor developed an elegant theory out of it. One may be tempted to believe at this point that all infinite sets are of equal size (they can be paired off with each other) but this is actually not the case. George Cantor showed by his brilliant theorem that infinities are of different sizes. In particular, he demonstrated that the set of natural numbers is of smaller size that the set of real numbers (their members cannot be paired off with each other). Furthermore, he showed that the set of real number is smaller than yet another infinite set (the power set of the set of real numbers), which is in turn smaller than yet another infinite set, and so on to infinity. Cantor also showed that one can measure the sizes of infinite sets in terms of new numbers, the so called transfinite cardinal numbers. Cantor believed that if all the transfinite cardinal numbers are taken together, they do not form a set anymore. They form what he called 'absolute infinity', an infinity whose size cannot be measured in terms of set theory. This absolute infinity Cantor believed to be God.


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