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Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the platonic realm of sets. The platonic realm of sets is supposed to be the world of truly existing sets, it is satisfaction in that world that we'd label as set theoretic truth. Now, the question is about existence of big sets, like the universe, Frege cardinals, Frege ordinals, the set of all of those, etc.. From the viewpoint of $\sf ZF$ existence of such entities is refuted altogether. However, what make us feel that $\sf ZFC$ is saying the truth of the whole set world in the platonic sense? I see the rules of $\sf ZF$ as being true of the Platonic realm of Well-founded sets, their motivation is the Cumulative Hierarchy, and the underpinning of its rules lies in the iterative conception of sets. The latter is most elegantly captured by the Cumulative Hierarchy. I don't see a clear motivation of it beyond this iterative picture. But, why should we think that the Platonic realm obeys such a trend. Why not restrict the rules of $\sf ZFC$ to the realm of well-founded sets, as it appears to be heavily ingrained in, and so permit existence of other kinds of sets with different rules.

For example if we have all rules of $\sf NFI$ to govern all sets, then simply add the schema of replacement restricted to the well-founded realm of sets, also add an axiom of existence of a set of all von Neumann naturals. Add an axiom of Choice over all sets. Also, to preserve some intuitive glimpse we add the axiom that all sets are strongly Cantorian.

$\forall X \exists f: f=\{ \langle x, \{x\} \rangle \mid x \in X \} $

And so we'd have the Category of sets being Cartesian closed!

All of those seems to be a way of welcoming the big sets to the platonic realm of sets with them having nice properties that we like. But, if there is a real world of sets, it need not obey these nice features. Actually they might defy Choice and Cartesian closedness. Now that $\sf Con(NF)$ proof of Randall Holmes has been Lean-verified. Then we might as well contemplate these big sets having awkward characteristics that defy basic intuitions about sets. And so instead of $\sf NFI$ we may take the whole of $\sf NF$ instead. And of course restrict Choice to the well-founded realm, and shun strong Cantorian axiom.

I chose $\sf NFI$ above because it is the strongest in comprehending over such sets in keeping with nice properties and intuitive expectations. $\sf NF$ may be an alternative because it is in some sense the maximal we have about comprehending over such sets.

So my question is about what really underpins our conception about existence of sets in the Platonic world should such a realm be there? Is it their formal mileage? Or is it some vague intuitive sense of them being real? To what extent we may accept a whole exotic world of sets lying there beyond the ardent cumulative hierarchy world of sets. What is the big picture here?

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    $\begingroup$ My understanding is that only part of the proof of Con(NF) has been verified, an important difficult part, but is it overstating matters to say we have a verified proof of Con(NF)? $\endgroup$ Apr 22 at 22:33
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    $\begingroup$ @JoelDavidHamkins, It's over. NF is consistent. The remaining part is only expositional for clarity and readability. The proof of existence of a structure for TTT has been verified, and that completes it. Practically this matter is closed!logicmatters.net/2024/04/21/nf-really-is-consistent, also see Randall Holmes home page. $\endgroup$ Apr 22 at 22:48
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    $\begingroup$ Although your question isn't about NF per se, it's worth pointing out that Randall Holmes himself does not regard the consistency of NF to have any philosophical consequences beyond what we can already get from the consistency of NFU, which was established decades ago. In this article, Holmes addresses some of the objections articulated by Joel David Hamkins, and tries to give an intuitive picture of what NFU is all about. $\endgroup$ Apr 23 at 2:38
  • $\begingroup$ @TimothyChow: Thanks for linking to that article, Timothy. I found it very enlightening and convincing (regarding its central thesis that ZFC has no special claim as the only possible foundation of mathematics)! And incidentally, while Holmes himself might not find the consistency of NF relevant to his arguments there, at a minimum it represents a technical upgrade over NFU from the perspective of those who are philosophically opposed to ur-elements, no? $\endgroup$ Apr 23 at 12:49
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    $\begingroup$ @TimothyChow, Sam Hopkins, well an Ur-element is understood not to be a set since they are discriminated from each other while having the same membership. If a theory is dominated by Ur-elements, like with the usual models of NFU in which you have more Ur-elements than sets, then naturally some people would object to it being a theory about sets. Well at least on the face of it, it is not solely a theory about sets. The presence of Ur-elements enables nice features like choice, it is easy to interpret relative to ZFC, more flexible technically, etc.. That's why many would prefer it. $\endgroup$ Apr 23 at 13:57

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The recent developments on the consistency of NF bring welcome closure to the longstanding open question about whether NF was consistent. And this is naturally a very important matter for those who find NF set theory attractive.

To my way of thinking, however, NF was never competitive in providing an attractive account of set theory, and the consistency question doesn't really change this. From my perspective, NF set theory is not based on expressing the fundamental truths of a coherent conception of the nature of set, as in the case of Zermelo's theory, but rather arises by a formal limitation of the axioms of an earlier naive (and easily-proved wrong) set theory so as to avoid a known proof of inconsistency.

I see NF as arising by a process of formal manipulation of the axioms, rather than a mathematical idea, and in my view this is not a sound method of finding robust meaningful principles in the foundations of mathematics. One doesn't reliably arrive at important or even interesting principles of set theory by fiddling with formal details like that.

The NF consistency question was open so long in large part because nobody had a coherent idea of what the theory was supposed to be about. There was little reason to think it was consistent or inconsistent, and the fact that it turns out to be consistent is something like a lucky accident. Perhaps the consistency proof will provide a picture of how we can conceive of the NF conception of sets, but I find this backwards and I believe that it will have little to do with our ideas about sets. NF is about something else.

Some philosophers of set theory sometimes express the view that the (inconsistent) general comprehension principle expresses something important, and they regret that it was shown inconsistent by Russell, seeking instead to save it somehow. This is what NF aims at, to save general comprehension as much as possible. This is also what motivates paraconsistent set theory.

My view, however, is that the general comprehension principle is simply a logical fallacy, one that is very easily shown to be false. It has a one-line refutation. It is wrong, and trivially so, and there is nothing there that needs to be saved. In this sense, general comprehension is something like denying the antecedent—an intuitively plausible principle for those who consider it naively, but ultimately it is shown wrong as a logical principle for trivial reasons. We don't mount huge foundational efforts to "save" the principle of denying the antecedent, and I see little reason to mount similar efforts to save the similar easily-proved-wrong principle of general comprehension, including the versions of it in NF.

Meanwhile, the Zermelo theory does express the expected fundamental truths of an understanding of sets arising in a transfinite cumulative hierarchy. The axioms are amply justified by the picture of sets arising in a vast cumulative hierarchy, where we may begin with some urelements, or none since they are not actually need for any purpose, and then form sets of them, and sets of these sets, and so on transfinitely. This picture of the set-theoretic universe leads easily to the Zermelo-Fraenkel theory, while incidently providing no support whatsoever for the general comprehension principle (although it does support the separation axiom, denied by NF). Also, it provides no support for the kind of sets that you describe outside the well-founded framework.

So I am inclined to take ZFC as expressing the fundamental truths of our understanding of sets as they arise in the cumulative universe of sets.

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    $\begingroup$ I'm not sure. Of course Russell also was primarily a philosopher, but I think his theorem refuting general comprehension is received very well in mathematics. Also Kripke is primarily a philosophy, but Kripke-Platek set theory is very well received. $\endgroup$ Apr 23 at 0:02
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    $\begingroup$ Neither ZC or ZFC arose from fundamental reasoning about sets as objects in the cumulative hierarchy. Most of the axioms of ZFC has their roots in Cantor and Bolzano's axiomatic endeavors, but they were naively presented. Zermelo's work was actually a way to avoid paradoxes, and not about the cumulative hierarchy "CMH". Actually CMH came later on. Zermelo just wanted to salvage Cantor's axioms, by putting it in a logical framework as to avoid paradoxes, he also added Choice to prove the well ordering theorem. NF didn't arise merely as a syntactical trick to salvage naïve comprehension. .cont $\endgroup$ Apr 23 at 0:02
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    $\begingroup$ @ZuhairAl-Johar Your historical points seem highly debatable, but my view is not about such historical contingencies. ZFC is seen today as fulfilling the cumulative hiearchy vision of sets. But there is no comparable idea underlying NF. NF does not seem to be about any particular understanding of sets, but is a theory arising from a purely formal consideration. $\endgroup$ Apr 23 at 0:08
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    $\begingroup$ @JoelDavidHamkins, Yes. I agree, but my question is why the cumulative hierarchy itself should dominate the Platonic world. CMH is just the well-founded set world nothing else. And if ZFC is about CMH, so let it be restricted to it, why should we stretch those rules to what may be beyond CMH. And why there should not be any beyond. There is no comparable motivation because the world ZFC is speaking about is EASY to ponder about. The big sets world is something hard. There is the idea of invariance under set like permutations of the universe. But this is not easy. $\endgroup$ Apr 23 at 0:11
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    $\begingroup$ To me, it is like asking why not allow transfinite numbers, so that we may deny any number being a successor of a number? Natural numbers are easy, but these transfinite numbers are hard. Also, NF is not just a mere formal trick, it has its origins in type theory. $\endgroup$ Apr 23 at 0:23
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Holmes: Hamkins' remarks here are important and I'd like to comment on them in detail to make my own position and goals clear.

Hamkins: The recent developments on the consistency of NF bring welcome closure to the longstanding open question about whether NF was consistent. And this is naturally a very important matter for those who find NF set theory attractive.

Holmes: I agree -- it should be noted that I have long been on record as not finding NF set theory attractive.

Hamkins: To my way of thinking, however, NF was never competitive in providing an attractive account of set theory, and the consistency question doesn't really change this. From my perspective, NF set theory is not based on expressing the fundamental truths of a coherent conception of the nature of set, as in the case of Zermelo's theory, but rather arises by a formal limitation of the axioms of an earlier naive (and easily-proved wrong) set theory so as to avoid a known proof of inconsistency.

Holmes: The historical origins of a theory do not always indicate what it really is as a theory. The original presentation of Zermelo theory is actually in various ways quite unsatisfactory because of the fact that its unifying conception had not yet been fully formed.

Hamkins: I see NF as arising by a process of formal manipulation of the axioms, rather than a mathematical idea, and in my view this is not a sound method of finding robust meaningful principles in the foundations of mathematics. One doesn't reliably arrive at important or even interesting principles of set theory by fiddling with formal details like that.

Holmes: Quine's own view seems to have been the contrary of this (he seems to speak in favor of fiddling with axioms until they work). But I am not speaking for him. There is an alternative story: Quine was motivated by a formal feature of the simple typed theory of sets which really is rather interesting and looks as if it might be a feature of the world: it is extremely polymorphic. It is also pretty clearly the case that Quine made a mistake in the original New Foundations paper in proposing strong extensionality: he talks about this decision, and in the course of that discussion he makes an actual mathematical mistake. NF is ill-posed; NFU should have been the proposal, and if it had been the history might have been a bit different.

Holmes, further: The history of NFU is an extension of the kind of history Hamkins deplores here: it is historically a correction of the system NF which appears to have been proposed simply because Jensen could see how to prove its consistency. But NFU does present a view of the world, which can both be seen in Jensen's construction of a model, and discovered by reasoning in NFU itself: the world of NFU can naturally be understood to be an initial segment of the cumulative hierarchy with an external automorphism. This view of the world is readily mutually interpretable with the view expressed in the Zermelo style theories.

Hamkins: The NF consistency question was open so long in large part because nobody had a coherent idea of what the theory was supposed to be about. There was little reason to think it was consistent or inconsistent, and the fact that it turns out to be consistent is something like a lucky accident. Perhaps the consistency proof will provide a picture of how we can conceive of the NF conception of sets, but I find this backwards and I believe that it will have little to do with our ideas about sets. NF is about something else.

Holmes: I agree! The initial entering wedge for showing that NF is consistent was defining tangled type theory, which gave a picture of a kind of world which this theory could be talking about (which is insane, by the way). My comment at the time was that upon seeing tangled type theory as equiconsistent with NF, I no longer had an opinion as to whether it was consistent or not. The actual proof of consistency is insanely delicately balanced: Wilshaw has the same impression. It just barely works, in many places. There might be some interest in exploring what other models of NF there may be, because I make very special assumptions to build mine and it is quite unclear what the range of possibilities is.

Holmes, further: but the case of NFU remains to be considered. Its history is just as unsatisfactory, but both external and internal further consideration of NFU lead to an actual picture of the set theoretical world (and not a terribly alien one).

Hamkins: Some philosophers of set theory sometimes express the view that the (inconsistent) general comprehension principle expresses something important, and they regret that it was shown inconsistent by Russell, seeking instead to save it somehow. This is what NF aims at, to save general comprehension as much as possible. This is also what motivates paraconsistent set theory.

Holmes: I so agree with you about this. The paradoxes are mistakes. In exposition of NFU, it can be useful to review them and point out how the fallacious arguments break down, but they play no essential role in mathematics.

I don't think Quine's motive actually was to save naive set theory in an ad hoc way, though you will find him saying this because he was a pragmatist about set theoretical axioms. He was already considering the simple typed theory of sets, which avoids all problems, is not in the least ad hoc, and admits a picture of the world. What he wanted to do was reduce ontological clutter: the world of the simple typed theory of sets looks like a hall of mirrors, and he wanted to collapse all types which appeared to look the same into one thing. And this does actually work -- very neatly if you allow urelements, and with insane difficulty in extensional type theory.

Hamkins: My view, however, is that the general comprehension principle is simply a logical fallacy, one that is very easily shown to be false. It has a one-line refutation. It is wrong, and trivially so, and there is nothing there that needs to be saved. In this sense, general comprehension is something like denying the antecedent—an intuitively plausible principle for those who consider it naively, but ultimately it is shown wrong as a logical principle for trivial reasons. We don't mount huge foundational efforts to "save" the principle of denying the antecedent, and I see little reason to mount similar efforts to save the similar easily-proved-wrong principle of general comprehension, including the versions of it in NF.

Holmes: I so agree, and I have conversations about this regularly with people with philosophical axes to grind.

Hamkins: Meanwhile, the Zermelo theory does express the expected fundamental truths of an understanding of sets arising in a transfinite cumulative hierarchy. The axioms are amply justified by the picture of sets arising in a vast cumulative hierarchy, where we may begin with some urelements, or none since they are not actually need for any purpose, and then form sets of them, and sets of these sets, and so on transfinitely. This picture of the set-theoretic universe leads easily to the Zermelo-Fraenkel theory, while incidently providing no support whatsoever for the general comprehension principle (although it does support the separation axiom, denied by NF). Also, it provides no support for the kind of sets that you describe outside the well-founded framework.

So I am inclined to take ZFC as expressing the fundamental truths of our understanding of sets as they arise in the cumulative universe of sets.

Holmes: I agree that the picture of the universe provided by the axioms of ZFC is very clear and compelling (though it does lead to questions if examined carefully). It is fun and interesting [and militates against some other things you say here] that NFU, the actually well-posed version of Quine's proposal, also leads to a coherent picture of the world...the same one, with a peculiar twist. The early history of Zermelo set theory is not as different from that of NF as you suggest here, but it fared better, in being cleaned up fairly soon (by Zermelo himself, among others) to give what is now the dominant theory. It is worth knowing that NFU admits a very similar cleanup. I comment here that NFU with additional useful principles is a serviceable foundation for mathematics and admits a coherent motivation. However, I also think Zermelo style foundations are a bit easier to work with. The fact that there is at least one alternative which also works is a useful thing to know when we think about what foundations do for us; it is not any part of my purpose to undertake a polemic to change the framework in which mathematics is done in practice, but it is important to know that there are other paths which could be followed.

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    $\begingroup$ +1. I am very glad that you have responded to my post, which I had worried was too critical, and I am very pleasantly surprised to find so many points of agreement. $\endgroup$ May 6 at 15:42
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    $\begingroup$ @JoelDavidHamkins There are some points of disagreement, too, please note. But there is a hazard both for you and for other witnesses to this conversation of ascribing to me beliefs and purposes which I do not have. That is a lot of the reason that I posted the longer reply. $\endgroup$ May 6 at 15:51
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    $\begingroup$ Yes, of course. I am glad that you have greatly clarified the nature of your views. $\endgroup$ May 6 at 15:53
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    $\begingroup$ The history of Quine's proposal of NF needs analysis. He made mistakes in the course of it, but it was much less ad hoc than it appeared, and Specker's 1962 results and Jensen's 1969 results, if they had been digested properly, show that the proposal actually did make sense. But NF itself is to say the least an ill-posed proposal. $\endgroup$ May 6 at 16:06
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    $\begingroup$ I do think that fascinating questions remain about NF, and I hope that someone answers them! The fact that an ill-posed proposal works does tell us that there is an unexpected interesting corner of the world which merits investigation. That is not the same as thinking we should all work in NF :-) $\endgroup$ May 6 at 16:22
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I find that I have a further comment in a different spirit on this paragraph.

Hamkins: Some philosophers of set theory sometimes express the view that the (inconsistent) general comprehension principle expresses something important, and they regret that it was shown inconsistent by Russell, seeking instead to save it somehow. This is what NF aims at, to save general comprehension as much as possible. This is also what motivates paraconsistent set theory.

Holmes: In general I agree with these remarks about "fiddling with the axioms to avoid the paradoxes". But there is another point to be made.

The simple typed theory of sets (type 0 with individuals, type 1 sets of individuals, type 2 sets of sets of individuals) is about something clearly describable (a baby version of the cumulative hierarchy, as it were). Its axioms look exactly like those of naive set theory except that one has to be sure that variables are typed correctly.

An alternative story about naive set theory (which is largely but not entirely appropriate to the actual facts of the early mathematical uses of naive set theory) is that naive set theory was an incorrect attempt to formalize a body of mathematical work which should have been done in simple typed theory of sets. On this narrative, there was something correct to be rescued: it wasn't that naive set theory was itself valuable at all, it was that the actual working theory in the background had not been expressed properly.

What is definitely true is that all the early foundational work can be done in the simple typed theory of sets (possibly with additional strong axioms of infinity) [with adjustments]. Historically, the simple typed theory of sets in the pure form needed to see the possibility of this formulation of foundations (and of the collapse of the types proposed by Quine, seen as a correction of simple type theory to reduce clutter, not as a correction of naive set theory) was not actually clearly presented until the 1930s, not long before Quine made his proposal.

If mathematical work has actually been done in a framework which turns out to be inconsistent, it is reasonable to look at the framework and see if it can be corrected to support the body of work which has been done...and the reason to think that this might be possible is not that the inconsistent framework has any value but the belief that the body of work actually done is of value and the framework may have been set up incorrectly.

Here is a horrible narrative which I do NOT think will actually happen but which one can contemplate as a thought experiment. Suppose that we have all been missing the relatively straightforward proof that every uncountable strong limit cardinal is singular (applicable to the proper class ordinal as well as to sets...) Then ZFC would be inconsistent. Would mathematics presented as founded on ZFC collapse? Absolutely not. Zermelo set theory with more modest strong axioms of infinity would be just fine, most likely, and would support most mathematics that has been done. (This isn't an NF-iste's fantasy: it would be just as uncomfortable on this view of the world, as it would prove that Henson's very reasonable Axiom of Cantorian Sets was inconsistent [which is roughly equivalent to having n-Mahlos for each n]). It would not be a reasonable response that all of mathematics [currently formalized in ZFC] would be invalidated: one would follow the mathematics to see what framework was actually needed for the correct work.

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  • $\begingroup$ Related is Frege's work on formalizing arithmetic, a work most of it can alternatively be seen to be based on Hume's principle. This is the basis for the neologism. Also, ZFC itself is nothing but a similar salvage of Bolzano-Cantor axioms which was presented within a general context of naïve approach to sets. $\endgroup$ May 15 at 20:35
  • $\begingroup$ The neologicist stuff is very weak. There is a much better way to salvage Frege's work using stratification, which gives a far stronger theory, closer to the intended strength. $\endgroup$ May 17 at 14:19

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