What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]

This question has been moved to philosophy.stackexchange.com

I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by axioms of: Separation schema, Collection schema, Set Union, Power set, and Infinity; can be said to be true of? This theory permits all kinds of abhorrent violation of extensionality and foundation, so what kind of entities suits such a theory? For example David Lewis in his book (Parts of Classes) had given an interpretation for classes as mereological aggregates of atomic labels of mereological aggregates. Now though he himself had postulated the principle that identifies parts of class as subclasses of classes, yet still in principle we can think of weaker labels that violate extensionality, i.e. we can think of having two distinct labels for the same aggregate, also labels can explain all kinds of circular membership. So it seems that atomic mereology + a singleton labeling function, can provide some intuitive envisioning into that fragment.

Question: had there been comparatively similar ideas that can explain intuitively such flagrant violations of Extensionality and Foundation?

• I find your question hard to understand, but maybe you'd be interested in mereology? – arsmath May 25 '18 at 6:46
• why I'm just asking about if the intuitive side of the above well known theory had been worked up before? I mean when Dana Scott presented his proof about such models, others as well, had then been works on the intuitive side of those entities? there should be some? – Zuhair Al-Johar May 25 '18 at 7:19
• Please post this kind of question here: philosophy.stackexchange.com – Monroe Eskew May 25 '18 at 12:27
• The problem is not clarity. It's that it's not a mathematical question. – Monroe Eskew May 25 '18 at 14:02
• No, I'm sorry, this is definitely NOT a mathematical matter. It is about philosophy of math. – Monroe Eskew May 25 '18 at 14:43

1 Answer

The axioms that you listed are satisfied in the cumulative hierarchy over any set of atoms. That is, begin with some entities that are not sets, for example people. Then consider (1) all sets of those entities, (2) all sets whose elements are among your original non-sets or among the sets formed in (1), and so forth, transfinitely, forming at level $\alpha$ all sets whose members are among your original non-sets or are sets from levels $<\alpha$. (A rather straightforward variant of ZF, called ZFA, describes such a universe. It is widely used for independence proofs concerning the axiom of choice.) Note that the original non-sets violate the axiom of extensionality, since they, like the empty set, have no members.

So, as far as the axioms you listed are concerned, any non-sets can be the non-extensional elements of your universe.

• I take seriously the absence of an axiom from a list of axioms in a theory, I mean as far as figuring out what is the intuitive entity underlying it because I hold a theory to capture all very basic properties of the underlying intuitive entity. Absence of an axiom must signal something, for example, the ZFA construction you've mentioned, begs an axiom of extensionality for sets (the non-empty objects), but this is NOT axiomatized here. Here in this theory even multiple copies of non empty sets is allowed, so the intuitive entity manipulated here is 'weaker' than what you are saying. – Zuhair Al-Johar May 25 '18 at 6:08
• the last line is not correct, since non-extensional elements of the universe might be non-empty and it would be really difficult to accept this statement as an intuitive qualification about them. I think the last line is meant to be about empty non-extensional entities, which is just one possible model of that theory. – Zuhair Al-Johar May 25 '18 at 9:14
• @ZuhairAl-Johar Note that I didn't say (in this answer) that sets can't be among the non-extensional elements. I said only that all non-sets can be among them. But in fact, in situations where there are violations of extensionality by things that have members, I'd be reluctant to call those things sets, precisely because of the violation of extensionality. Typical examples would be multi-sets and lists. – Andreas Blass May 25 '18 at 11:27
• still I don't see a clear answer really. what are those entities that those axioms are speaking about? they can't be just qualified as any non-sets, for example suppose I have two distinct non empty entities k,l both having the same members, these are non-extensional entities, now if I say that those could be ANYTHING THAT IS NOT A SET, then this is weird, for example I take that to be they could be humans for example? Perhaps I mis-understood what you are saying, perhaps you mean I can put anything that is non-set in the universe of discourse of this theory, yes, but that is not helpful. – Zuhair Al-Johar May 25 '18 at 11:39
• @ZuhairAl-Johar Your last two comments look reasonable, and they don't conflict with anything I actually wrote. In particular, I did not say that non-sets that have members could be anything that is not a set, e.g., humans. I said that anything that is not a set (e.g., humans) can be in a ZFA universe and that it would not have members. – Andreas Blass May 25 '18 at 12:42