Timeline for Incompleteness and nonstandard models of arithmetic
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2021 at 4:55 | comment | added | Vladimir Reshetnikov | @MarcAlcobéGarcía An example of an arithmetic statement (that I find almost obviously intuitively true) about generalized multi-level polynomials that is not provable in PA, but provable in ZFC (it is quite similar to Goodstein’s theorem): math.stackexchange.com/q/1371535/19661 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 2, 2010 at 14:28 | comment | added | François G. Dorais | For example, the Paris-Harrington Theorem - en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem - and Goodstein's Theorem - en.wikipedia.org/wiki/Goodstein%27s_theorem | |
| Jun 2, 2010 at 14:07 | comment | added | Marc Alcobé García | Another question would be then if ZFC proves any mathematically interesting arithmetic statement (other than Con(PA) or some Gödel sentence for PA) that PA cannot prove. | |
| Jun 1, 2010 at 19:56 | vote | accept | Marc Alcobé García | ||
| Jun 1, 2010 at 19:41 | vote | accept | Marc Alcobé García | ||
| Jun 1, 2010 at 19:56 | |||||
| Jun 1, 2010 at 16:23 | comment | added | François G. Dorais | To clarify, moving to ZFC does reject some nonstandard models, but not all such models. For example, since ZFC proves Con(PA), no model of PA + ¬Con(PA) can be interpreted as the omega of a model of ZFC. | |
| Jun 1, 2010 at 15:17 | history | edited | François G. Dorais | CC BY-SA 2.5 |
clarification
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| Jun 1, 2010 at 14:56 | comment | added | François G. Dorais | No, moving to ZFC doesn't help. This is what the second paragraph is about: no matter what theory you decide to interpret arithmetic in, if there is a model at all then there must be one with nonstandard integers. Perhaps surprisingly, this is not related to incompleteness per se, those are just properties of first-order logic. | |
| Jun 1, 2010 at 14:48 | comment | added | Marc Alcobé García | Although not stated clearly, the idea is not to get rid of every nonstandard model (nor of every countable one), which, as you mention, is impossible. I would be happy if one could get rid of one of them (which could be impossible too, certainly I don't know). Restating the second part of the question: would moving to a stronger theory such as ZFC remove any of the nonstandard models of the weaker one (PA)? Moving from Q to PA certainly does! (See Smith's notes for an example). And what's more important. Does this have any relation with Incompleteness? | |
| Jun 1, 2010 at 12:51 | history | edited | François G. Dorais | CC BY-SA 2.5 |
minor addition
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| Jun 1, 2010 at 12:16 | vote | accept | Marc Alcobé García | ||
| Jun 1, 2010 at 12:16 | |||||
| Jun 1, 2010 at 12:00 | history | edited | François G. Dorais | CC BY-SA 2.5 |
correction
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| Jun 1, 2010 at 11:35 | history | answered | François G. Dorais | CC BY-SA 2.5 |