Timeline for The smallest initial ordinal which is not defined using first-order formulas with parameters of smaller ordinals?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2017 at 16:54 | comment | added | Zetapology | Ah, ok. I see now. | |
| Oct 12, 2017 at 14:41 | comment | added | Noah Schweber | Now this might seem a problem if $M$ itself is pointwise definable, but it's not: just because $x$ is undefinable in $(N, \in)$ doesn't mean it's undefinable in $(M,\in)$! A definition of $x$ in $M$ may well exist "higher" than $N$ - e.g. $x$ may be coded into the continuum pattern in $M$ above $\kappa$. | |
| Oct 12, 2017 at 14:39 | comment | added | Noah Schweber | @Zetapology Let me sketch why Emil's claim is true. Suppose $M$ is a model of ZFC with an inaccessible $\kappa$; I claim that in $M$, there is a set model of ZFC which is not pointwise definable (and - by the same argument - whose $\theta_0$ exists, and which has $\theta$-fixed points, and etc.). Specifically, think about $N:=V_\kappa^M$. $M$ proves that $N$ is uncountable, and can also define definability for set-sized structures, so $M$ can run the "math tea argument" to show that $N$ has an undefinable subset - that is, there is some $x\subset N$, $x\in M$ which is undefinable in $(N,\in$). | |
| Oct 12, 2017 at 14:35 | comment | added | Noah Schweber | @Zetapology That's not what Emil said. Emil said that the consistency strength of the existence of $\theta_0$ (and more) is strictly below that of an inaccessible cardinal, which is true. He didn't claim that an inaccessible cardinal proves the existence of $\theta_0$. | |
| Oct 12, 2017 at 14:33 | comment | added | Zetapology | Well, @EmilJeřábek, like Noah said, just because the existence of $\theta_0$ doesn't imply an inaccessible cardinal, doesn't mean it has low consistency strength. In fact, $V=HOD$ should not be compatible with an Inaccessible if it was below it, as he stated. | |
| Oct 12, 2017 at 11:01 | comment | added | Noah Schweber | @EmilJeřábek Yes, that's of course true. Not my finest moment. :P | |
| Oct 12, 2017 at 10:48 | comment | added | Emil Jeřábek | Unless I am missing something, if $M$ is a well-founded model of ZFC whose ordinal height has uncountable cofinality, then there does exist a (least) fixpoint of $\theta$ in $M$. Thus, the “consistency strength” of the existence of such a model is fairly low (below an inaccessible cardinal). | |
| Oct 12, 2017 at 5:35 | comment | added | Noah Schweber | @Zetapology "Wow... I guess this cardinal is simply a really strong, yet somehow small type of cardinal." No, the point is that the hypothesis "we are not in a pointwise definable model" is really orthogonal to the usual large cardinal hierarchy. There's no reason yet to believe that the existence of (say) a $\theta$-fixed point cardinal with cofinality $\omega$ should have high consistency strength over ZFC; it's just not outright implied by any of the standard large cardinal axioms (namely, those compatible with V=HOD). | |
| Oct 12, 2017 at 4:19 | comment | added | Zetapology | In fact, if $M$ is a transitive inner model of ZFC such that for every first order $\phi$ and every $v$ a finite sequence of ordinals in $M$, $\phi(v)\Leftrightarrow M\models\phi(v)$ (i.e. $M$ is almost "complete" in a sense), then the critical point of an elementary embedding $j:V\rightarrow M$ is always a theta fixed point. | |
| Oct 12, 2017 at 4:06 | comment | added | Zetapology | Does the existence of a critical point of some nontrivial elementary embedding from $j:\mathrm{HOD}\rightarrow\mathrm{HOD}$ imply the existence of at least $\theta_0$? I know every Reinhardt cardinal is a $\theta$-Fixed point. | |
| Oct 12, 2017 at 4:00 | comment | added | Zetapology | Wow... I guess this cardinal is simply a really strong, yet somehow small type of cardinal. | |
| Oct 12, 2017 at 3:26 | comment | added | Noah Schweber | @Zetapology See Theorem 4 and Observation 5 of this paper on pointwise definable models. If $P$ is a large cardinal compatible with V=HOD (and this includes basically everything), then we can have pointwise definable models realizing $P$ - and in a pointwise definable model, even $\theta_0$ doesn't exist. So you won't get such a simple sufficient condition. (That paper is well worth a read by the way, especially given your interests.) | |
| Oct 12, 2017 at 3:10 | comment | added | Zetapology | So this does solve the second question in the comments, but the question of what axiomatic requirements are necessary to warrant the existence of $\Theta_0$ are there; for example, does the existence of a weakly inaccessible imply the existence of $\Theta_0$? I still accept this question and commend you for your answer however. | |
| Oct 12, 2017 at 3:08 | vote | accept | Zetapology | ||
| Oct 12, 2017 at 3:07 | vote | accept | Zetapology | ||
| Oct 12, 2017 at 3:08 | |||||
| Oct 12, 2017 at 2:33 | history | answered | Noah Schweber | CC BY-SA 3.0 |