Set theory without infinite sets

Question

Consider a theory of sets in which there is no infinite set. One may argue that a robust theory of this kind would mimic the real world more accurately and would be more applicable. Why are such theories not prevalent?

Answer 1

I don't find the examples in the comments persuasive. Maybe we can do without the real numbers; I never liked them anyway. But here is a different sort of example that I think might be more compelling.

Consider a finite computer program whose input is a finite string of symbols and whose output, when it has one, is "yes" or "no" depending on whether the string has some property. Just which property isn't important right now. But I am not thinking of anything tricky. It could be something like "the string has even length" or "the string is a number that is a multiple of 23" or "the string describes a graph that has a Hamiltonian cycle", where we would normally imagine that the property was perfectly well-defined, and clear at least in principle.

The set of strings for which the algorithm says "yes" is infinite, and I cannot think of any way in which a finite set could be said to describe the situation more accurately or realistically. What finite set would you use? You would be in the position of saying that there was some finite constant $\mathcal L$ such that the algorithm would fail to say "yes" for every string longer than $\mathcal L$. But what $\mathcal L$ would you pick?

I suppose you could argue that any real implementation of the algorithm on a real computer would take more than the lifetime of the universe to finish if given a string bigger than, say, $10^{10^{10}}$ symbols, which is certainly true, but what does that add to the understanding of the algorithm? All it does is drag in a lot of contingent facts about current engineering practice and currently-accepted theories of physics. But the whole point of studying algorithms is to abstract away exactly such contingent facts, and to model computation in a simpler way. We are only trying to model a certain computation, not the entire universe, so why do we need to include the speed of light and the cosmological constant in our model?

That said, there is a legitimate school of thought in mathematics that holds that idea of arbitrarily large integers is incoherent. You should read about ultrafinitism. You may find it interesting.

Answer 2

If you take the theory ZF and replace the Axiom of Infinity with its negation, you get a theory of sets in which there must not be any infinite sets.

It turns out that this theory is equivalent to first-order Peano Arithmetic! The paper here has a good explanation.

Obviously PA is very well studied!