Timeline for Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 17, 2019 at 21:34 | history | edited | Elliot Glazer | CC BY-SA 4.0 |
typo fix
|
| Aug 17, 2019 at 21:08 | vote | accept | Adam Epstein | ||
| Aug 16, 2019 at 21:00 | history | edited | Elliot Glazer | CC BY-SA 4.0 |
Sketching a model I emailed the author, per his request.
|
| Aug 11, 2019 at 0:12 | comment | added | Adam Epstein | This might be a good place to post your sketch (which I would duly accept) though there might well be better places. | |
| Aug 6, 2019 at 11:12 | comment | added | Elliot Glazer | The infinitely many chains might be extraneous. They're an artifact from a previous model. There should be models of ZC+TC without definable infinite sets. I'll email you a sketch I have in mind. | |
| Aug 5, 2019 at 22:32 | comment | added | Adam Epstein | As it turns out, I did post something, just nowhere accessible. Typical :) Cheers. mathoverflow.net/questions/140969/… | |
| Aug 5, 2019 at 21:32 | comment | added | Adam Epstein | I think the main difference between your approach and mine is that you are implicitly working in ZFA rather than ZF, since you presuppose a $\mathbb{Z}$ or $\mathbb{Z}^2$ worth of objects which you wish to treat as urelements, so that you don't have to worry about seeing them arise again, inconveniently, after a round of adjoining powersets. Of course, ZF and ZFA are equiconsistent, and model-theoretic (and more general mathematical) practice is more consonant with ZFA anyhow. | |
| Aug 5, 2019 at 20:58 | comment | added | Adam Epstein | Presenting things this way seems to clarify the otherwise fiddly details one needs to attend to in verifying the axioms of Z, by putting it into a setting closer to that of Matthias. I'd be happy to send you slides of the most recent version I've been speaking about. Postscript to my "education" comment, I have no clue about my original question for Z+set-induction (let alone Z+Transitive Containment) -what seemed a promising idea turned out to be nonsense - and I'd be very interested in any thoughts you might have. | |
| Aug 5, 2019 at 20:49 | comment | added | Adam Epstein | and in fact, the various meanings of hereditary finiteness in the absence of Foundation, Transitive Closure, or both, is part of where my research led). It thereby suffices to prove the consistency of this theory. One way to proceed is by the iterated adjunction of powersets. Another, which I find cleaner, is to (definably, again) perform a Rieger-Bernays twist to the membership relation. Either way, one obtains a model of Z-Foundation which satisfies the additional axiom, which one may then (definably) cut back to the sets whose support in $\mathfrak{S}$ is bounded from below. | |
| Aug 5, 2019 at 20:37 | comment | added | Adam Epstein | This past year I've been reorganizing my presentation as follows. Adjoin to Z an additional axiom $\mathfrak{S}\neq\emptyset \wedge \forall x\exists n\in\omega \bigcup^n x \subseteq \mathfrak{S}$ where $\mathfrak{S}$ is the class of hereditary singletons (and where $\omega$ may well be a proper class); note that this is a first-order assertion in the language of set theory. Then, much as you observe, the universe admits a definable automorphism which fixes only the elements of $V_\omega$ (the hereditarily finite sets, but this terminology clashes with my other use of "hereditary" | |
| Aug 5, 2019 at 20:22 | comment | added | Adam Epstein | I do think one needs to be a bit careful about specifying this adjusted $\mathcal P$ operation, since one must coherently perform the indicated replacements at every stage of the infinite process But it's straightforward to formulate and prove a lemma about extending an appropriate relational structure by an adjusted $\mathcal P$ operation, and then passing to the direct limit over $\omega$-many iterations. | |
| Aug 5, 2019 at 20:19 | comment | added | Adam Epstein | possibly losing Foundation, of course, but when starting from a model built out of "hereditary singletons" one can cut back to the part whose "support" is bounded from below. Actually, what you write here mathoverflow.net/questions/314483/… is closer, since there you have only one chain of hereditary singletons. In your post above, you seem to have infinitely many such chains: is there a particular reason for this? | |
| Aug 5, 2019 at 19:58 | comment | added | Adam Epstein | Hi, Eliot. Yes, one can build models of Z in this way. I arrived at a similar construction a few years after my post, but except for alluding to it here later on mathoverflow.net/questions/201718/… I never posted my own solution here. I did write up some notes, including the fiddly details about why the resulting structure is a model of Z. More generally, starting from any $\omega$-model of finite set theory (Z-Infinity+$\neg$Infinity), one may adjoin iterated powersets in a similar external fashion, | |
| Aug 5, 2019 at 18:55 | vote | accept | Adam Epstein | ||
| Aug 5, 2019 at 19:21 | |||||
| Jun 21, 2019 at 5:07 | history | answered | Elliot Glazer | CC BY-SA 4.0 |