Timeline for Existential statement without witness
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 17, 2014 at 9:58 | vote | accept | McDuffin | ||
| Dec 16, 2014 at 22:33 | comment | added | Christoph-Simon Senjak | @Emil Jeřábek: Sorry, yes. It is an example for "for every numeral $\tilde{n}$ we have $PA\models\Theta(n)$" but $PA\not\models\forall n\Theta(n)$, got that mixed up. | |
| Dec 16, 2014 at 17:22 | comment | added | Emil Jeřábek | @Christoph-SimonSenjak: Not really. Goodstein’s theorem is $\Pi^0_2$, so it does not even start with an existential quantifier. And if you substitute a number for the outer universal quantifier to obtain an existential ($\Sigma^0_1$) statement, it will have a witness provable already in Robinson’s $Q$. | |
| Dec 15, 2014 at 23:18 | comment | added | Christoph-Simon Senjak | Goodstiein's Theorem (Kirby-Paris-Theorem) is an example for PA. | |
| Dec 14, 2014 at 18:06 | comment | added | Asaf Karagila♦ | "Shape without form, shade without colour, Paralysed force, gesture without motion" -- T.S. Eliot, The Hollow Men. | |
| Dec 14, 2014 at 17:41 | comment | added | Gerald Edgar | I believe Goedel was the first to give such an example. (His construction works for any sufficiently expressive system, in particular for ZFC.) | |
| Dec 14, 2014 at 16:52 | answer | added | Emil Jeřábek | timeline score: 10 | |
| Dec 14, 2014 at 15:31 | answer | added | Joel David Hamkins | timeline score: 23 | |
| Dec 14, 2014 at 15:22 | comment | added | McDuffin | I meant an arithmetic statement, so ZFC $\vdash \exists x \in \omega \phi(x)$ | |
| Dec 14, 2014 at 15:19 | review | First posts | |||
| Dec 14, 2014 at 15:21 | |||||
| Dec 14, 2014 at 15:16 | history | asked | McDuffin | CC BY-SA 3.0 |