There are particular traps in thinking about axioms for sets. Perhaps the most general is that it is difficult to reason about sets without using our ordinary reasoning faculties, which include ideas about sets. When we discuss axioms it is necessary to keep your attention on what is implied by the axioms in distinction to what is true about sets. Another trap is that without a little introduction to formal logic it is not easy to know what kinds of reasoning are to be allowed lest we sneak in, or even appear to sneak in conventional set ideas. Indeed some relative consistency theorems use sophisticated mathematics.

I will not here try to elucidate formal logic but instead rely on that part of informal logic that corresponds to predicate calculus. I will use “collection” for the ordinary idea of set but bear in mind that some particular collections that you may know, may not have set counterparts that are forced to exist by whatever collection of axioms are on the table at some moment.

I want to talk about the power set axiom which has led me into some confusion in years past. It says that for any set S there is another set P with the property that any element of P is a subset of S and that any set that is a subset of S is an element of P. For years I read this as saying that any sub collection of S is a set which belongs to P. If we let S = {0, 1, 2, 3} I could well omit {0, 2, 3} from sethood and thus not be required to include it in P. It turns out that other axioms force me to admit {0, 2, 3} as a set, and therefore membership in P, but not by the power set axiom alone. These other axioms ensure that all of the finite sub collections are indeed subsets. These axioms provide for all sorts of infinite subsets but can never “get them all”.

In the above I imagine a game between me with the goal of finding the most outrageous model for set theory and you who provides axioms that constrain me to reasonable and useful models. It is not clear that Euclid had such a concept but modern mathematics has adopted such a search for outrageous models as a fruitful way of discovering what may not be proven by certain axioms as well as what may be.

The power set axiom fails to ensure that there is a set whose members are 0, 2 & 3. It merely requires that if there should be such a set that it belong to the power set. Other axioms do the work of requiring the existence of the set {0, 2, 3}. But we might imagine other subsets for which the other axioms are too feeble. I had imagined that the power set axiom required the existence of all those sub collections as subsets and furthermore that they be collected together into one set, called the power set. The axiom does only the latter! The only set that the axiom creates is the power set proper, but it can’t populate it! It creates the club, other axioms must furnish the members!

This sheds light on the axiom of choice. Some cannot imagine that the axiom is not true. The purpose of the axiom might be said to outlaw certain models of sets that seem not to lead to fruitful mathematics. Some mathematicians feel to the contrary that the AC is unjustified as a manner of argument. Brouwer, who made the most impassioned attack on the axiom, is most famous for his earlier work where he required the axiom.

Gödel pointed out that any set of consistent axioms has a countable model. This includes the axioms leading to integers and subsets of integers. The set of subsets of integers is, in ordinary logic, uncountable. How can this be? The proof of uncountability of the integer subsets is by Cantor and uses arguments that are perfectly valid in elementary set theory. Cantor’s proof merely proves that there is no function that provides a correspondence between integers and subsets of integers. A function in this theory is in fact a set and it would seem that there is indeed by Cantor’s argument, no set providing the correspondence. The correspondence proven by Gödel’s argument is not a set whose existence can be proven within the axiomatic set theory. Indeed Gödel’s proof is not constructive and cannot be used to construct the set within the axiom set. The axioms do not and cannot say everything we believe about sets.

Skolem, T. “Sur la portée du théorème de Löwenheim-Skolem.” Les Entretiens de Zurich sur les fondements et la méthode des sciences mathématiques (December 6-9, 1938), pp. 25-52, 1941.
See this note for a good intro to Löwenheim-Skolem, and then my note on Gödel’s incompleteness theorem.