What is the intuitive notion that ZF minus Extensionality minus Regularity plus Collection capture?

Question

In order to clarify my questions I'll here introduce the concept of intuitive completeness of an axiomatic system, an axiomatic set A of a consistent theory would be said to be intuitively complete if for some intuitive notion I all very basic aspects of I are grasped by A . For example ZFC seem's to be complete with respect to the intuitive notion "Set", basically because it contains the axioms of Extensionality and (to some authors) Foundation. Had it missed these axioms, then it would have been deemed intuitively incomplete with respect to the intuitive notion of 'set'.

Now it is known that the following axiomatic system, a fragment of the axiomatic system of ZFC, is equi-interpretable with the whole of ZFC:

1. Set Union
2. Power
3. Separation
4. Collection
5. Infinity

Definitely this system is not intuitively complete with respect to intuitive notion 'set'.

This fragment is technically interesting, simply because most of mathematics can be interpreted in it, hence this question about its intuitive completeness.

Question: is there a known intuitive notion I that the above axiomatic system is intuitively complete with respect to it?

Dana Scott had investigated the above system, proved its equi-interpretability with ZFC, others did similar work with other standard theories, but I don't see an intuitive explication of the entities in those theories, in the sense of intuitive completeness mentioned above.

Question: Had that matter been explored before?

One thing to be mentioned is that since the axiomatization is a fragment of ZFC, a theory about sets, then whatever that intuitive account about those entities is, it must be part of the intuitive account about sets, in other words the intuitive notion I that the above system would be considered as intuitively complete of, must be "weaker" than the intuitive notion of "set". It must not conflict with the notion of set, and definitely need not be stronger than it.

Presenting models of this theory in which some form of extensionality hold like models of ZFA and trying to figure out the intuitive notion related to them won't be helpful in figuring out what I is, since they would be clearly a stronger notion than I and the axiomatic system above would be clearly intuitively incomplete with respect to it, since that weak form of Extensionality must be added to the above list of axiom for latter to be intuitively complete of, this is unhelpful much as presenting a model of this theory that is a model of ZFC is unhelpful. We need an intuitive notion that just fits these axioms.

My personal attempt is that I think that those entities might be "collectivity states of affairs", those need not be extensional, nor well founded, nor choice respective. The same set of objects can be collected by distinct collectivity states of affairs, the latter refers to a process of collectivity of the collection, for example two distinct persons might collect the same collection, but the states of affairs of collectivity of that collection are distinct, having different details, etc.. so for that axiomatic system the "is a member of" intuition given to symbol epsilon is to be replaced with "is collected by", we see that collectivity need not respect foundation, since self-collectivity is a possibility, it can be cyclical, it can go to infinity. Choice is the main axiom that gives the impression of being about collectivity rather than collection, since it carries the impression of an act, still imagining non-choice respective collectivity is justified. Collectivity seems to be a weaker concept than a set, since the later is a collectivity state of affairs, but fairly specific and rigid one, one that is completely determined by what is collected. Even the cumulative hierarchy of sets, seems to be about collectivity rather than the passive set concept. So all in all the notion of collectivity state of affairs seems to fit this axiomatic system that the latter is judge to be intuitively complete of!

Question: had there been a research on a topic that is similar to what I'm suggesting here, I mean the collectivity state of affairs.

Answer 1

In Computer Science, the simplest interpretation of this would be a class of multi-sets, containing references to const multi-sets. But they could also have other properties. I think traditionally though a common reason to deny ordinary extensionality is to produce something like ZFA in a one-sorted universe. This can be done by limiting non-extensionality to collectivities without elements and equating those to atoms². So you might intuitively interpret this system as a class of references to multisets that can contain atoms.

Edit: It is possible this is too strong and specific, so you might want to check it over. This is an attempt at the most simple and conventional intuition on how to interpret a system of exactly those five axioms. The key point I believe in excluding extensionality is that it allows multiple versions of sets, which by itself seems to suggest multisets. Other than that, excluding choice and foundation while weakening extension can produce atoms as in ZFA per references. I'm not sure "is collected by" is distinguishing in the universe itself without additional definitions given collection and separation. If you were to implement a subset of the class though it would be.

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2: cf. section 4.1 Weak extensionality and ZFA. Kanamori, A., Gabbay, D. and Woods, J. (2012). Sets and Extensions in the Twentieth Century, Volume 6 (Handbook of the history of logic). Amsterdam: Elsevier, pp.582-583.