Timeline for Is diamond consistent with 2nd order PA?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 6, 2023 at 14:00 | comment | added | Joel David Hamkins | Your title question doesn't seem to match the questions you ask in the body of the post, since in the title you ask for consistency (which is obviously yes) and in the body you ask for provability. | |
| Dec 6, 2023 at 13:38 | answer | added | Joel David Hamkins | timeline score: 2 | |
| Dec 6, 2023 at 8:48 | history | edited | YCor |
edited tags
|
|
| Nov 24, 2023 at 10:18 | history | edited | YCor |
edited tags
|
|
| May 22, 2023 at 22:22 | answer | added | Hanul Jeon | timeline score: 4 | |
| May 14, 2023 at 19:30 | comment | added | Vladimir Kanovei | I may add that some ensuing results do not go through in PA2 and/or ZFC$^-$, like eg Jensen's theorem 1970 on minimal $\Pi^1_2$ real singleton - neither via diamond nor via Jensen's original argument (which also involves ordinals bigger than $\omega_1$). | |
| May 14, 2023 at 19:21 | comment | added | Vladimir Kanovei | Ali: yes as a shceme. That is, fix a canonical $\Delta_1$ definable diamond-sequence by say Jech-millenium, this is a formula say A of PA2. Say immediately that A defines a (coded) $\omega_1$ sequence of ctble sets of ordinals. Then claim that for any formula B which defines a subclass of $\omega_1$ the class $G$ of ordinals with right guesses is stationary. And stationarity needs another $\forall$ quantifier over formulas. | |
| May 13, 2023 at 21:46 | comment | added | Ali Enayat | Do you mean third order arithmetic instead of second order arithmetic? I don't see how $\diamond_{\omega_1}$ can be stated in the language of second order arithmetic (unless you think of it as a scheme). | |
| May 12, 2023 at 20:02 | comment | added | Hanul Jeon | I guess the proof of $\diamondsuit$ presented in Jech's book shows $\mathsf{GBC}^-+\diamondsuit_{\omega_1}$ has the consistency strength below $\mathsf{KM}^-$, by carrying over Jech's argument over $\mathsf{ZFC}^-$ + $V=L$ + "$L_{\omega_1}$ exists", whose consistency strength is equal to that of $\mathsf{KM}^-$. (I guess $\mathsf{GBC}^-$ + $\Pi^1_1$-Separation + $\Sigma^1_1$-Class Choice might be enough.) | |
| May 12, 2023 at 19:32 | history | edited | YCor |
edited tags
|
|
| May 11, 2023 at 20:27 | history | asked | Vladimir Kanovei | CC BY-SA 4.0 |