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Can we claim that there are infinite number of objects when the set of the objects does not exist?

For example, there is no set of all sets, but can we still say that there are infinitely many sets (of any kind)? And what would that mean? How would we say that formally?

One way is just to say: look, there are infinitely many subsets of integers, so, of course, there are infinitely many sets of any kind overall. But saying that, we must rely on something like "a superset of an infinite set is infinite" or something similar. And what would that mean to say that there are infinitely many sets?

Another example is from here: provide-different-proofs-for-the-following-equality. The set of all proofs of a given theorem is not defined, but we can clearly describe an infinite set of such proofs (which is NOT a subset of the set of all proofs because such a set does not exist). Can we still claim that there are infinitely many proofs of the theorem, and what would that claim mean exactly?

P.S. If you are aware of any literature discussing this or similar questions, you may just guide me through the literature by providing some references. I would definitely appreciate it.

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There are infinitely many sets if one can produce a collection of sets that cannot be bijected with an element of $\omega$. $\omega$ itself is one such collection. Also, the set of all proofs of a given theorem is defined, so long as you specify the proof system you are working in (say, the Hilbert system). It is easy to see that there are infinitely many proofs of a theorem, just by tacking on tautological statements in the middle, and that just means that the set of all proofs of that theorem cannot be bijected with an element in $\omega$. There's no need to philosophize here... – Isaac Solomon Apr 15 '12 at 18:13
I think it depends on what you mean by the "infinite number of objects", if you think about an ordinary number I suppose it does not make a sense because ordinary numbers are sets by definition. – dawid Apr 15 '12 at 18:13
The set of all proofs of a given theorem does exist! – Michael Greinecker Apr 15 '12 at 18:17
What does it mean for a mathematical object to exist? – Asaf Karagila Apr 15 '12 at 18:22
The set of all sets does not exist, but the class of all sets does exist, and it's infinite. See – Jim Belk Apr 15 '12 at 18:25

Let change a bit the question. What does mean finite ?

There is several definition. One of them is that a set is finite if it has an injection in a finite ordinal (a finite ordinal is an element of the first limit ordinal). Of course, one can extend this definition to collections (a collection is a predicate with one free variable) of set. And it is clear that if a collection has a injection into a finite ordinal, then the collection is itself a set. Indeed, form the injection, ou can construct a bijection onto a smaller ordinal, and then the collection is the image of the ordinal under the inverse map, hence it is a set (replacement axiom).

So if a collection is not a set, it is not finite in this sense.

Another definition for a set (or collection) to be finite is that every injection of the collection into itself is surjective. For sets, and with dependent choice axiom, this is equivalent to the former definition. Things are a bit more complicated, but I think the conclusion should be the same. I can see a proof with the foundation axiom : let C be a collection, finite in that sense. Then every set $C\cap V_\alpha$ is finite. Consider the non-decreasing map $\alpha \mapsto \operatorname{card} C\cap V_\alpha$, form ordinals to finite ordinals. This map is eventually constant, so there exists an $\alpha$ such that $C\cap V_\alpha = C$. Thus, $C$ is a set.

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Euclid proved there are an infinite number of primes without ever knowing set theory. He formalized it as follows: no (finite) list of primes contains all primes. You can apply similar formalization to all of your examples, showing that no finite list of set contains all sets of no finite set of proofs contains all proofs.

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But is there a set of all primes? Couple of years ago I spoke with a finitist who told me that he does not believe that the "set of all even numbers" exist, but you can still talk about properties common to "all the even numbers". Your answer, while incomplete, hits this very spot that Jim Belk's comment to the question hit. You need not for the set to exist, you just care about the collection. Of course for someone unfamiliar with the distinctions this just as well be syntactic sugar... – Asaf Karagila Apr 17 '12 at 20:06

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