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

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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?

## closed as unclear what you're asking by Gro-Tsen, Peter LeFanu Lumsdaine, Piotr Hajlasz, arsmath, Monroe EskewMay 25 '18 at 12:20

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• 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

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.