Timeline for Is ZF equivalent to Specification + Sentence Reflection?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2022 at 9:17 | comment | added | Zuhair Al-Johar | @NoahSchweber, You mean: working in ZFC, for any ordinal $\kappa$ satisfying $\forall T \ [\exists \alpha \, (V_\alpha \models T) \implies \exists \beta < \kappa \, ( V_\beta \models T)]$, the set $V_\lambda$, for $\lambda > \kappa$, would be a model of this theory. | |
| Mar 10, 2022 at 21:10 | comment | added | Noah Schweber | @ZuhairAl-Johar There are only set-many (continuum-many to be precise) theories, without parameters, in the language of set theory. | |
| Mar 10, 2022 at 21:02 | comment | added | Zuhair Al-Johar | @NoahSchweber, why should such $\kappa$ exist? | |
| Mar 10, 2022 at 19:57 | comment | added | Noah Schweber | @ZuhairAl-Johar Think about what happens if $\kappa$ is large enough that every theory satisfied by some level of the $V$-hierarchy is satisfied by some level of the $V$-hierarchy below $\kappa$. Then any level of the $V$-hierarchy above $\kappa$ will satisfy "parameter-free reflection" for a silly reason. | |
| Mar 10, 2022 at 19:56 | comment | added | Zuhair Al-Johar | @NoahSchweber, what do you exactly mean by sufficiently large? | |
| Mar 10, 2022 at 19:53 | comment | added | Zuhair Al-Johar | $ U=V_\alpha$ is defined in the customary manner, that's why I didn't mention it. $\alpha$ is an ordinal, i.e. an $\in$-well founded transitive set of transitive sets, or equivalently an $\in$-well ordered transitive set. $V_\alpha$ is the union of the range of a function from $\alpha+1$ having the image of the successor being the power of the image of its predicessor, and at limit stages the union of the images of prior ordinals. That's the usual meaning of $V_\alpha$ | |
| Mar 10, 2022 at 19:18 | comment | added | Gro-Tsen | (Just to belabor the point, if $\psi(\alpha,U)$ is the formula which says “lots of ZFC axioms are true, and also $U = V_\alpha$”, then in ZFC it's equivalent to $U = V_\alpha$, and clearly “$\exists \alpha: \exists U: \psi(\alpha,U) \land \varphi^U$” is going to state much more than you intended.) | |
| Mar 10, 2022 at 19:15 | comment | added | Gro-Tsen | How is the formula $U = V_\alpha$ expressed? (Or maybe you just need to express $\exists\alpha\in\mathit{On} : U = V_\alpha$, perhaps that's easier.) I mean, I know what it means in ZFC, but I'm not convinced that there aren't multiple versions which would be equivalent in ZFC but not in the axiom system you're proposing. (Also, it matters to the elegance of the system being proposed: it seems nice, but not if there's a lot of complexity hidden in that $V_\alpha$.) | |
| Mar 10, 2022 at 18:19 | comment | added | Noah Schweber | If you don't allow parameters in your reflection scheme, I don't think this has a hope of working: consider $V_{\kappa+\omega}$ for sufficiently large $\kappa$. | |
| Mar 10, 2022 at 18:14 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |