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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.

  • "Intuitive completeness" is very unclear. What is an intuitive notion, what is a "very basic aspect," and what is it to "grasp" such a thing? – Monroe Eskew May 25 '18 at 18:23
  • I think "intuition" is not easy to define rigorously, here it is related to analysis of informal meanings, for example it is intuitive to say that sets are extensional, one can instinctively know, or even feel that, or as I state 'grasp' it. Yes these are not rigorous terms, but one can figure out what it means though. For example the intuitive picture behind ZFA doesn't fit the above system, it is stronger than it, this can be almost 'sensed' intuitively. I think though one can indeed find the place of defining intuition in analysis of informally presented meanings. – Zuhair May 25 '18 at 21:50
  • I have the impression that most of the work done these days on intensional equality is done in the context of some flavor of type theory. – Hurkyl May 25 '18 at 23:35
  • This is murky. One person's intuition isn't someone else's. And ZF being intuitively complete? ZF's conception of sets is completely contrary to anyone's intuitive notion of sets (unless one's intuition comes from studying ZF). First, unrestricted comprehension fails. So a set is NOT "a collection of things defined by a predicate." Secondly, the axiom of infinity leads to a lot of trouble. All kinds of strangeness come out of assuming infinite sets exist. If there's anything ZF isn't, it's intuitive. One has to be trained to believe that ZF captures an intuitive notion of what a set is. – user4894 May 26 '18 at 3:21
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    @user4894, I still hold (though not insistent) that a set is an extension of a predicate, it is an object illustrating a predicate, it mirrors predication by membership, so 'set' is a discriminative function on predicates much as number is a discriminative function on cardinality of sets, with number it is bijection that ensures the discrimination, with set it is the prediation-membership paralleling that is doing the trick. I don't think there is something wrong with this. That it failed for the general case of predicates doesn't mean it is wrong, it just mean that the domain of the – Zuhair May 26 '18 at 8:53
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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.


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.

  • I think this explanation is too specific, and strong. I like the word "collectivities" those seem to be weaker than "collections" as I mentioned in the main posting. – Zuhair May 25 '18 at 21:49
  • If you add references this would help your answer and give me something to check. For example, do you have a reference claiming "the common reason to deny ordinary extensionality is to produce something like ZFA in a one-sorted universe"? – Frank Hubeny May 25 '18 at 22:19
  • @Frank changed to "a common reason" and added a reference – Greg S May 25 '18 at 23:38
  • Yes this multi-set +atoms and other violations of foundation and choice can present models of this theory, but this is not the point. The point is at intuitive level what are those entities, I saw in multi-sets there is an interpretation of them in terms of sets, which is what I don't like, since intuitively speaking a multi-set is a weaker notion than a set, so it must cover wider range of entities more than just sets, to go interpret them in terms of a special sector of sets, is going in the opposite direction. as I said is like defining water in term of oceans. – Zuhair May 26 '18 at 14:45

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