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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...2012-04-15
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    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.2012-04-15
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    The set of all proofs of a given theorem *does* exist!2012-04-15
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    What does it mean for a mathematical object to exist?2012-04-15
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    The set of all sets does not exist, but the *class* of all sets does exist, and it's infinite. See http://en.wikipedia.org/wiki/Class_(set_theory).2012-04-15
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    @Asaf: I wish I'd see that question asked more often :-)2012-04-15
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    @AsafKaragila: Is there something subtle about existence here I'm not getting? You [seem](http://math.stackexchange.com/questions/46833/how-do-we-know-an-aleph-1-exists-at-all/46836#46836) to have a good idea what it means for $\aleph_1$ to exist.2012-04-15
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    @Michael: Yes, existence can be taken as a Platonist existence of objects or as a formal statement as "provable from the following axioms..." it is not clear whether the OP makes this distinction or even aware of it.2012-04-15

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