Timeline for Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails?
Current License: CC BY-SA 2.5
7 events
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| Sep 8, 2010 at 23:48 | comment | added | Andrés E. Caicedo | ... if you find it too dry, I recommend that you take a look at the Handbook of Proof Theory. I believe a different proof can be found there. | |
| Sep 8, 2010 at 23:47 | comment | added | Andrés E. Caicedo | Hi David, sorry I couldn't reply earlier, it's been terribly busy here today. Anyway, funny thing, I was going to suggest that you look at precisely the book you found, Logic and Combinatorics (Contemporary Mathematics, vol 65, AMS, 1987). Last I checked, it was still available through the AMS. It is a lovely book, really. There are some technical articles, and some very nice expository ones. The final article, by Steve Simpson, is a great introduction to the area of unprovability and combinatorics. The article by Buchholz and Wainer requires some familiarity with proof theory... | |
| Sep 8, 2010 at 17:31 | comment | added | David E Speyer | Alright, I've now found a reference which does claim to prove the result you want. "Provably Computable Functions and the Fast Growing Hierarchy" books.google.com/… As yet, I can't understand the proof at all... | |
| Sep 8, 2010 at 13:43 | comment | added | David E Speyer | Andres: A question about your paper. You say that your Theorem 1.9 is the main result of Wainer, "A classification of the ordinal recursive functions". Yet Wainer's paper does not mention Peano arithmetic or induction restricted to $\Sigma_k$ formulas. How do I see Theorem 1.9 in Wainer's notation? (It doesn't help that Wainer doesn't define the "Grzcgorczyk hierarchy" or "Peter's $k$-recursive functions".) | |
| Sep 1, 2010 at 0:32 | comment | added | Jason DeVito - on hiatus | I didn't mean "natural" in any formal sense (and I can edit out the word or change it to a better one if you'd prefer). I meant "natural" in the sense of a model which could be used in a proof that Goodstein's theorem is independent of PA. I just needed some adjective to rule out models whose existence requires us to already know Goodstein's theorem is independent of PA (like the model guaranteed by Godel's completeness theorem applied to "PA + Goodstein's theorem is false"). | |
| Sep 1, 2010 at 0:30 | comment | added | Jason DeVito - on hiatus | Thank you for your response. I'll be sure you check out both your paper and the Kirby-Paris proof (I do have easy access to the paper). | |
| Sep 1, 2010 at 0:06 | history | answered | Andrés E. Caicedo | CC BY-SA 2.5 |