Timeline for Definitional complexity of truth in $L$ without $0^{\#}$
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 9, 2017 at 14:44 | vote | accept | Elliot Glazer | ||
| May 9, 2017 at 2:42 | comment | added | Elliot Glazer | Let us continue this discussion in chat. | |
| May 9, 2017 at 2:39 | comment | added | Joel David Hamkins | One can definitely have a forcing invariant definition of the theory of $L$ in $V$ without $0^\sharp$, just by coding the theory into the GCH pattern arbitrarily high up. Alternatively, if there is a class club of $\kappa$ with $L_\kappa\prec L$, then you can make those cardinals the cardinals of $V$, and this way, even if you do more forcing, the theory of $L_{\omega_1^V}$ will be the theory of $L$, so the $\Sigma^1_\omega$ definition works, even when $0^\sharp$ does not exist. | |
| May 9, 2017 at 2:29 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 9, 2017 at 2:26 | comment | added | Joel David Hamkins | It's definitely not invariant under forcing, since we could collapse more cardinals, and this would change $\alpha$. So you are right, it can't be $\Sigma^1_3$. I guess it is $\Delta^1_4$. | |
| May 9, 2017 at 1:59 | comment | added | Elliot Glazer | ^ Actually I'm unsure about this. If your definition really is $\Sigma_3^1,$ then it is invariant under forcing right? That seems hard to believe. | |
| May 9, 2017 at 1:56 | comment | added | Elliot Glazer | Now it seems this definition would not be invariant under forcing. If we ask for a definition of truth in $L$ which is invariant under forcing, would that require $0^{\#}$ to exist? | |
| May 9, 2017 at 1:23 | comment | added | Joel David Hamkins | Ah, let's do it that way then. | |
| May 9, 2017 at 1:20 | comment | added | Elliot Glazer | Are you computing the complexity of $t \subset \omega$ or of $\{t\} \subset \mathcal{R}?$ In the case of the former, $\Pi_3^1$ would follow from $\Sigma_3^1$ simply by checking if $L \models \neg \varphi.$ | |
| May 9, 2017 at 1:05 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 9, 2017 at 0:54 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 9, 2017 at 0:45 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 9, 2017 at 0:30 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 9, 2017 at 0:15 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |