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The topic of this post was shifted to

https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist

Since it was deemed to be a philosophical question about sets.

In nutshell the question asked was the following:

If we hold that some platonic realm exists in which only sets exist, then is it natural to hold that big sets (like the set of all sets, Frege's cardinals and Frege's ordinals, etc..) would exist in that realm? And if NOT then why? That's to say if one holds that only extensional well-founded sets can exist in that platonic set realm, then what is the intuitive justification for such a restriction?

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  • $\begingroup$ There are a variety of conditions we might impose in order for something to qualify as an "object". The broadest is simply our being able to talk about it consistently; having a procedure that distinguishes it from other objects is a stronger requirement. I see these questions as largely definitional. $\endgroup$ – Nik Weaver Mar 3 '18 at 16:31
  • $\begingroup$ I didn't downvote, but maybe people would consider this question more appropriate for the FOM discussion list (although the quality of discourse there tends to be rather low). $\endgroup$ – Nik Weaver Mar 3 '18 at 16:40
  • $\begingroup$ @Nik Weaver but we can talk about big sets consistently, this is done in NFU. $\endgroup$ – Zuhair Al-Johar Mar 3 '18 at 17:26
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    $\begingroup$ Personally I don't think ZF is true of the world of all sets. $\endgroup$ – Nik Weaver Mar 3 '18 at 18:57
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    $\begingroup$ This is a philosophical, not a mathematical question. $\endgroup$ – Qfwfq Mar 3 '18 at 22:15
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Well-foundedness is part of what I mean by "set". (Extensionality is another part.) And ZFC summarizes some aspects of my intuitive conception of sets.

Certainly other sorts of things exist. For example, I exist, and I'm not a set. You ask when it's natural to hold that entities of certain sorts exist. I'd find that natural to the extent that I have a clear intuitive concept of those entities. And I find it natural to work with theories that describe such entities. For example, I have no clear intuitive picture of any entities for which I can see that the axioms of NF are true, so I don't find it natural to work in NF (even though the axioms of NF are nice and clean).

It is a rather remarkable empirical fact that, when I have a clear intuitive concept of some mathematical entities, then I can usually code those entities as sets, perhaps not perfectly but well enough for most mathematical purposes. The same seems to be true for other people's intuition as well, and it seems to be why most of us accept ZFC (perhaps plus some large cardinal axioms) as a reasonable foundation for mathematics.

I don't know whether this empirical fact is a psychological one (about my "clear intuitive concepts") or a historical one (about mathematical concepts that people happen to have introduced) or something else.

By the way, "perhaps not perfectly" above was intended to cover my doubts whether, for example, the standard set-theoretic view of the real line as a collection of points really matches the intuition of continuity underlying $\mathbb R$. It matches well enough, in the sense that we can do analysis using the set-theory version of $\mathbb R$, but I'm not sure that it really matches the full intuition.

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    $\begingroup$ +1 for "I'm not sure that it really matches the full intuition" $\endgroup$ – David Roberts Mar 4 '18 at 5:59
  • $\begingroup$ @ Andreas Blass, It is nice to hear those intuitions of yours about sets. I have a question: is that intuition of yours about well-foundedeness and extensionality being essential to the nature of sets is something related to the Qunian principle of "No entity without identity", alluded to in the article referred to in the abovementioned link? I mean that there is a clear way to check the identity of sets in terms of their elements in the pure well-founded realm. $\endgroup$ – Zuhair Al-Johar Mar 4 '18 at 12:15

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