Timeline for Is every true statement independent of $PA$ equivalent to some consistency statement?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 30 at 9:38 | comment | added | Kotlopou | @Wojowu It indeed isn't correct, as per this question: math.stackexchange.com/questions/5003237/… | |
| Feb 20, 2018 at 16:51 | comment | added | Simon Henry | By all strategy are winning strategy I mean't all recursive strategy. But you're right I went back to Kirby & Paris paper and it seems that you are right and my memories of it were largely oversimmplified. This being said what they proof made me doubt that Goodstein theorem is not stronger than the consistency of PA. | |
| Feb 20, 2018 at 16:30 | comment | added | Wojowu | @SimonHenry I'm afraid that can't be correct. Well-foundedness of $\varepsilon_0$ is a statement much stronger than Con(PA) (in particular, I believe it implies Con(PA+Con(PA))). Also, I don't think Goodstein's theorem (a single sentence) implies full Kirby-Paris theorem (which, formally, can't be even stated in the language of PA, but we can express it as a scheme like well-foundedness). I have recently discovered a reference to the fact that adding a single sentence can't increase the proof-theoretic ordinal, but I don't have it at hand. | |
| Feb 18, 2018 at 21:20 | vote | accept | Christopher King | ||
| Feb 18, 2018 at 16:05 | answer | added | Payam Seraji | timeline score: 8 | |
| Feb 18, 2018 at 9:54 | answer | added | user103227 | timeline score: 10 | |
| Feb 18, 2018 at 8:50 | comment | added | Wojowu | Do you have a reference for Goodstein being equivalent to Con(PA)? | |
| Feb 18, 2018 at 2:43 | history | edited | Christopher King | CC BY-SA 3.0 |
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| Feb 18, 2018 at 1:43 | review | Close votes | |||
| Feb 18, 2018 at 11:23 | |||||
| Feb 18, 2018 at 1:28 | comment | added | Andrés E. Caicedo | No, far from it. Consistency statements, in the sense you intend, are $\Pi^0_1$ statements. It would be more interesting to ask the version of your question where you add this additional syntactical requirement. | |
| Feb 18, 2018 at 0:44 | history | edited | YCor |
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| Feb 18, 2018 at 0:15 | history | asked | Christopher King | CC BY-SA 3.0 |