I found something at Vanderbilt College about the "Axiom of Choice". The author poses this problem: he wants to define "small" sets of positive integers. He wants the definition to including the following properties:

- Any set with zero or one members is "small".
- Any union of two "small" sets is "small".
- A set is "small" if and only if its complement isn't "small."

The "complement" of a set S means the set of positive integers not in S. I might mention that the sets of positive integers is uncountably infinite. If a proof is needed, here's a simple one:

Suppose we have sets of positive integers A_{1}, A_{2}, A_{3},... Let *n* be a positive integer. Either *n* is in A_{n} or it's not. Define N = {*n*:*n* is not in A_{n}}. N is a defined set given A_{1}, A_{2}, A_{3},... Then *n* is in N if and only if *n* is not in A_{n}. That means that N can't be A_{n} for any *n*.

Any alleged one-to-one mapping of the positive integers onto the sets of positive integers is defeated by the existence of another set outside of the list of sets. Getting a one-to-one mapping for all sets of integers is literally impossible. The sets of positive integers must be uncountable. As I commented in my previous post, things get squirrelly with the uncountable.

Back to the author's problem, it's important to disconnect our own notion of "small" with "small" as he would like to define it. If we keep to our intuitive notion of small, property a satisfies it. Property a and b together mean that any finite set must be "small". That includes a set with a googleplex of elements. But a googleplex is tiny compared with infinity. A finite set may or may not be intuitively small. But necessarily, it must be "small" if properties a and b are satisfied.

The author gives examples where any two of the three properties are satisfied:

- A "small" set is defined as a finite set. That fails to satisfy property c, because neither the evens nor the odds are "small".
- A "small" set is a set not containing the number 1. This satisfies properties b and c, but not a: {1} isn't small by this definition.
- A "small" set contains at most one of the first three numbers. This satisfies a and c, but fails b: {1} and {2} are small, but {1,2} isn't small.

The author's claim is that a definition exists, but it is impossible to come up with it as an example. Proofs of the existence are non-constructive, and cannot be made constructive. He does give an apparent summary of the proof, which was incomprehensible to me. I don't have the background for even the terminology, let alone the substance. But from what he says, the Axiom of Choice is required to prove the existence of a definition of "small" satisfying all three properties.

Right now, I'm still very uncertain about the Axiom of Choice. For this particular problem, I'd rather conclude that no such solution exists, rather than that such a solution exists but we can't find it even in approximation.

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