Timeline for Can we have external automorphisms over intersectional models?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 10 at 15:55 | comment | added | Zuhair Al-Johar | Thank you very much! | |
| May 10 at 15:54 | comment | added | Joel David Hamkins | Yes, exactly that. | |
| May 10 at 15:52 | comment | added | Zuhair Al-Johar | Ah! I see what you mean. But, let me phrase it differently, you mean $\bigcap_n \alpha_n$, lets call it $\kappa$, this would be an initial segment, $\kappa$ would be in $M$, then we choose any of the $\alpha_n$ sets, and take the subset of it $\alpha_n \setminus \kappa$, this would be an element of $M$ by intersectionality, and clearly this has no $\in$-minimal element, this mean that $\alpha_n$ itself can be seen to be non-well ordered inside $M$ itself. | |
| May 10 at 15:47 | comment | added | Joel David Hamkins | I meant it might not be a descending sequence of predecessors. But my argument in any case handles also that case. The limit of the descent will be in $M$ by intersectionality, and so $M$ will be able to see the illfounded nature. | |
| May 10 at 15:37 | comment | added | Joel David Hamkins | It is not the same, since the descending sequence might not be a descending $\omega$ sequence. The ordinal set $\lambda$ cannot be seen as a well-founded ordinal in $M$, since the intersection is a set in $M$, and the complement of that in $\lambda$ is a nonepmpty subset of $\lambda$ in $M$ with no minimal element. | |
| May 10 at 15:31 | comment | added | Zuhair Al-Johar | Yes, its the same one I gave, but why it cannot be seen inside $M$ as a well-founded ordinal? I mean externally it doesn't have an immediate successor, but why should that transfer inside $M$? | |
| May 10 at 15:27 | comment | added | Joel David Hamkins | I had mean $\bigcap_n\alpha_n$ where $\alpha_0>\alpha_1>\cdots$. The intersection will reveal an initial segment of the ordinal whose final segment has no least element .And so it cannot be seen as a well-founded ordinal inside $M$, which is the situation I was imagining. | |
| May 10 at 15:21 | comment | added | Zuhair Al-Johar | Let $\alpha$ be such one. You mean $\bigcap_{n \in \omega} (\alpha-n)$, where $\lambda - 1$ is the predecessor of $\lambda$ externally seen. Hmmm...., but that intersection can be pseudo-ordinal also, $M $ would see it as an ordinal, but why it cannot see a successor for it? | |
| May 10 at 15:06 | comment | added | Joel David Hamkins | I think it can't happen, since if the thing has strictly descending sequence in the ambient setting, then you can intersect the initial segments occuring on that sequence and remain inside $M$, which will be an ordinal in $M$ below it having no successor. | |
| May 10 at 15:02 | comment | added | Zuhair Al-Johar | Yes! Exactly. It is those that I call non-standard ordinals, or call them pseudo-ordinals, those are the ones moved by the ambient theory. Can we have such a model provided it is also intersectional as seen by the ambient theory? | |
| May 10 at 14:57 | comment | added | Joel David Hamkins | Or perhaps you mean to ask about the situation: can there be a standard model $\langle M,\in\rangle$ in a ZF-Reg setting, such that sets that are not ordinals in the ambient setting be seen as ordinals inside $M$? | |
| May 10 at 14:48 | comment | added | Zuhair Al-Johar | May be? Thanks for the reference. | |
| May 10 at 14:43 | comment | added | Joel David Hamkins | You are mixing up what it means to be a nonstandard model with nonstandard ordinals and a model of a theory without foundation. These are totally different. If the ambient theory has what it thinks are ordinals, then these will be fixed by any automorphism of $M$ available in that ambient setting, even if that setting is seen as nonstandard from a further metameta setting. | |
| May 10 at 14:41 | comment | added | Joel David Hamkins | There is a discussion of automorphisms of universes of set theory without foundation in the joint paper arxiv.org/abs/1311.0814. | |
| May 10 at 14:40 | comment | added | Zuhair Al-Johar | Those rank shifting automorphism do exist, but can the model be at the same time externally seen as being intersectional. | |
| May 10 at 14:38 | comment | added | Zuhair Al-Johar | Yes, I mean for model M (the one inside) this is in reality a non-well founded set, it has ordinals that are well founded and those would be fixed. But also it has sets that it sees as ordinals but from the outside they are not ordinals, from the outside those non-standard ordinals do have infinite descent with respect to ordinal $<$ relation, but from the inside this is not seen. The $V_\alpha$ stages are seen from inside M as such, but if $\alpha$ is non-standard externally then those are not real powersets. | |
| May 10 at 14:09 | comment | added | Joel David Hamkins | Boffa models have numerous automorphisms, but they all fix the well-founded sets, including the ordinals, and so they aren't rank shifting in that sense. I don't believe there is any satisfactory notion of rank outside the well-founded ordinals, and I'm not sure what you mean by "rank" if it didn't refer to a well-ordered hierarchy of ordinals. | |
| May 10 at 13:58 | comment | added | Zuhair Al-Johar | I mean suppose the inside model satisfy a theory with regularity. So, it has a rank notion inside it, but the model is non-well founded externally, what I need is for this model to admit rank shifting automorphism and be intersectional at the same time. Can this be cooked inside Boffa set theory? | |
| May 10 at 13:55 | vote | accept | Zuhair Al-Johar | ||
| May 10 at 13:41 | comment | added | Zuhair Al-Johar | I see. I wanted the model to be intersectional, admit external automorphism but I need that to be rank shifting, The rank is seen from inside the model as so, but from outside it is not so, its index is in reality a non-standard ordinal, i.e. a transitive set of transitive sets that has an infinitely descending membership subset. | |
| May 10 at 13:26 | comment | added | Joel David Hamkins | These models have only standard ordinals, which are fixed by every automorphism. Without Reg, in the general case, there is no satisfactory theory of "rank". That is why I find your ambient theory unsatisfactory as a foundational theory of sets. | |
| May 10 at 13:21 | comment | added | Zuhair Al-Johar | One important issue, can one of these automorphisms be rank shifting? That is, it can shift some non-standard ordinal $\alpha$ say inwardly? That is, we have $j(\alpha)< \alpha$, for an automorphism $j$? | |
| May 10 at 13:16 | comment | added | Joel David Hamkins | I edited to explain. I removed the Boffa angle, which is unimportant. | |
| May 10 at 13:16 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 156 characters in body
|
| May 10 at 13:02 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |