# Is it natural to hold that Ur-elements, small & big sets and proper classes exists? [closed]

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?

• 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. – Nik Weaver Mar 3 '18 at 16:31
• 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). – Nik Weaver Mar 3 '18 at 16:40
• @Nik Weaver but we can talk about big sets consistently, this is done in NFU. – Zuhair Al-Johar Mar 3 '18 at 17:26
• Personally I don't think ZF is true of the world of all sets. – Nik Weaver Mar 3 '18 at 18:57
• This is a philosophical, not a mathematical question. – Qfwfq Mar 3 '18 at 22:15

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.