Timeline for Can there be such an elementary embedding?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2018 at 19:46 | vote | accept | Zuhair Al-Johar | ||
| Jul 13, 2018 at 9:47 | comment | added | Zuhair Al-Johar | I do have a proof that the "j" in Boff construction need not be an automorphism. | |
| Jul 13, 2018 at 9:38 | comment | added | Zuhair Al-Johar | So if I replace "elementary embedding" with "automorphism" in my question, then would that circumvent your argument? | |
| Jul 13, 2018 at 6:44 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jul 13, 2018 at 6:07 | comment | added | Noah Schweber | @ZuhairAl-Johar The models in the Boffa construction are necessarily ill-founded, I believe, since we need a rank-moving automorphism. $\mathfrak{S}$ meanwhile is used in two different ways in my answer - before the fold and after the fold - and is defined separately each time. In the former it is a $\{\in\}$-structure with domain $M\sqcup P(M)$, and in the latter it is a $\{\subseteq\}$-structure with domain $P(M)$ | |
| Jul 13, 2018 at 5:20 | comment | added | Zuhair Al-Johar | still you didn't tell me what is $\mathfrak{A}$ | |
| Jul 13, 2018 at 5:03 | comment | added | Zuhair Al-Johar | the original models in Boffa construction are not assumed to be "ill founded" | |
| Jul 13, 2018 at 4:58 | comment | added | Zuhair Al-Johar | by the way the structure I'm speaking about is an $\{\in,j\}$ structure since j is allowed to be used in both separation. | |
| Jul 13, 2018 at 4:58 | comment | added | Noah Schweber | @ZuhairAl-Johar No, it's not affected by that (although with the assumption that $M$ is transitive, we get to avoid a bit of circumlocution, which is nice). Re: Boffa models, unless I'm missing something the automorphism $j$ of the illfounded model $W$ you use to construct the model is not an elementary embedding from $(V_{\alpha+1})^W$ (where $\alpha$ is a nonstandard ordinal appropriately moved) into itself. | |
| Jul 13, 2018 at 4:55 | comment | added | Zuhair Al-Johar | OK guilty as charged. I'll write that. But anyhow your argument doesn't seem to be affected by that, isn't it? | |
| Jul 13, 2018 at 4:54 | comment | added | Noah Schweber | @ZuhairAl-Johar And where precisely did you state that in your question? | |
| Jul 13, 2018 at 4:53 | comment | added | Zuhair Al-Johar | Yes, $M$ is transitive. | |
| Jul 13, 2018 at 4:52 | comment | added | Noah Schweber | @ZuhairAl-Johar $M$ is not a subset of $P(M)$ unless $M$ is transitive ... | |
| Jul 13, 2018 at 4:48 | comment | added | Zuhair Al-Johar | first: what is $\mathfrak{A}$? Now: $M$ is the domain of a model of $ZF$, $M$ is both a subset and an element of $P(M)$, $j$ is an automorphism from $P(M)$ to $P(M)$ that moves subsets of $M$ that are definable in the language of set theory (with parameters in $M$ or without any parameters) in $M$. You say that this is not possible, but by then how I am to understand Boffa models that uses automorphism that moves a rank internally? those are used in the proof of $NFU$, in those models you do have $j(M)$ being an element of $M$, see math.boisestate.edu/~holmes/preserves3.pdf (page 5). | |
| Jul 12, 2018 at 23:26 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jul 12, 2018 at 23:16 | history | answered | Noah Schweber | CC BY-SA 4.0 |