Timeline for Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 26 at 0:12 | vote | accept | Arvid Samuelsson | ||
| Jul 12 at 10:44 | comment | added | Farmer S | ...unique, so one could alternatively argue that that isomorphism is also in $L_\beta[G]$, and hence also in $L_\beta$.) | |
| Jul 12 at 10:44 | comment | added | Farmer S | @HanulJeon Because $N$ models $T$, which includes "$V=L[0^\sharp]$", it is naturally coded by a subset $N'$ of $\alpha$. And the height of $N$ is exactly $\alpha$, so since we have a code for $\alpha$ in $L_\beta[G]$, we can (with a $\Sigma_1$ formula) ask for both a code $N''$ for a model, and an isomorphism between $\alpha$ and the ordinals of $N''$. But $N'$ is the unique such code, so $N'\in L_\beta[G]$, so $N'\in L_\beta$. This gives $N\in L_\beta$, since, as you say, $N$ is externally wellfounded. (Actually the isomorphism between the ordinals exhibited by $N''$, with $\alpha$, is also.. | |
| Jul 11 at 8:34 | comment | added | Hanul Jeon | @FarmerS It sounds like you identified $N$ with a set of ordinals coding $N$, which is possible as $L_\beta[G]$ has a global well-order. Is it what you intended? Also, how can we recover the transitive model from the coding inside $L_\beta$? We know $L_\beta$ models $\mathsf{KP}$, but recovering a transitive collapse from a well-founded relation is impossible just from $\mathsf{KP}$. Do we need the coding to be well-founded externally (i.e., not only over $L_\beta$ but also over $V$) to recover the transitive collapse over $L_\beta$? | |
| Jul 11 at 8:19 | comment | added | Farmer S | ...Show there is some $p\in G_1$ which decides the truth of "$\tau\in\sigma_1$" for every name $\tau$.) | |
| Jul 11 at 8:17 | comment | added | Farmer S | @HanulJeon It's Solovay's theorem that if $W\models$ ZFC and $W$ is transitive and $\mathbb{P}\in W$ and $A$ is a set of ordinals in $W[G]$ for every $(W,\mathbb{P})$-generic filter $G$, then $A\in W$. The same proof works here, replacing the collection of all $W$-generics with "sufficiently generic (over $L_\beta$)", which we can take to mean generic and such that $L_\beta[G]\models$ KP. (Here is a sketch of the proof: Take $(G_1,G_2)$ to be mutually generic with $G_1,G_2$ each sufficiently generic, and let $\sigma_1,\sigma_2\in L_\beta$ be names for $N$ w.r.t. $G_1,G_2$ respectively... | |
| Jul 11 at 6:34 | comment | added | Hanul Jeon | @FarmerS Why does $N\in L_\beta[G]$ for a sufficiently generic $G$ imply $N\in L_\beta$? | |
| Jul 9 at 15:09 | answer | added | Arvid Samuelsson | timeline score: 4 | |
| Jul 9 at 13:50 | comment | added | Farmer S | $N\in L_\beta$ because it is $\Delta_1(\{x\})$ uniformly in codes $x$ for $\alpha$. So it gets into $L_\beta[G]$ whenever $G$ is sufficiently $(L_\beta,\mathrm{Coll}(\omega,\alpha))$-generic, and in fact is in $L_\beta$ already. | |
| Jul 9 at 13:46 | comment | added | Arvid Samuelsson | I'm primarily interested in the statement in bold, and your proof above has the same problem: why does $N \in L_\beta$? I'll try to study the answer of the other question you linked to see if that answers my question. | |
| Jul 9 at 6:15 | comment | added | Farmer S | There is also more related material in the question/answers mathoverflow.net/questions/418801/can-local-0-exists-in-l | |
| Jul 9 at 6:00 | comment | added | Farmer S | ...For suppose $N$ is the unique such model. Then $N\in L_\beta$ where $\beta$ is the least admissible $>\alpha$. But $(0^\sharp)^N\notin L^N=L_\alpha$, so there is an ordinal $\gamma\in[\alpha,\beta)$ such that $L_\gamma$ projects to $\omega$. So there is a surjection $\pi:\omega\to\alpha$ with $\pi\in L_\beta$ (in fact $\pi\in L_{\gamma+1}$). But every set in $\mathcal{P}(\alpha) \cap L_\beta$ is definable from parameters over $N$, so $\pi$ is definable from parameters over $N$, contradicting that $N$ models ZFC. | |
| Jul 9 at 6:00 | comment | added | Farmer S | For (i): Let $\alpha$ be the least ordinal such that there is a transitive model $M$ of ZFC + "$0^\sharp$ exists" of ordinal height $\alpha$. (I am assuming such an ordinal $\alpha$ exists; so $\alpha$ is also countable.) Then there is also a model $N$ of height $\alpha$ satisfying the theory $T$, where $T$ is ZFC + "$0^\sharp$ exists" + $V=L[0^\sharp]$. Any two distinct models of $T$ of height $\alpha$ are ($\subseteq$)-incomparable (and in fact also incomparable in the sense of iterating)... | |
| Jul 9 at 5:27 | comment | added | Farmer S | You didn't actually ask a question. (It seems there are two possible implied questions: (i) why does said theory have incomparable minimal models?, and (ii) why is the boldface statement true? Are you asking one and/or both of them (or something else)?) | |
| Jul 8 at 20:57 | history | asked | Arvid Samuelsson | CC BY-SA 4.0 |