Timeline for How many models of Peano arithmetic are isomorphic to the standard model and how many models of Peano arithmetic are non-standard?
Current License: CC BY-SA 3.0
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| Dec 18, 2023 at 14:11 | history | edited | Joel David Hamkins |
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| Mar 25, 2012 at 15:44 | comment | added | Jason Rute | Samuel, as for your question about how many isomorphic models of the standard model there are, it doesn't have a very useful answer. The simple answer is that there is as many as there are sets (in say ZFC), which is to say the class of standard models of PA is a proper class. For each set $A$ just consider the model of pairs $(n,A)$ where each $n$ is a standard natural number (properly coded). With an appropriate multiplication and addition intepretation this is a model of PA. That is why we usually count models (or algebraic structures) up to isomorphism only. | |
| Mar 24, 2012 at 23:02 | vote | accept | Samuel Reid | ||
| Mar 24, 2012 at 21:16 | answer | added | Joel David Hamkins | timeline score: 17 | |
| Mar 24, 2012 at 20:14 | comment | added | Andrés E. Caicedo | Two excellent references: 1. "Models of Peano Arithmetic" by R. Kaye. (In particular, the specific fact you ask for and several variants are discussed there.) 2. "The structure of models of Peano Arithmetic", by R. Kossak and J. Schmerl. | |
| Mar 24, 2012 at 20:01 | answer | added | Andreas Blass | timeline score: 25 | |
| Mar 24, 2012 at 19:34 | comment | added | Guillaume Brunerie | "How many models of Peano arithmetic are there that are isomorphic to the standard model?" One, up to isomorphism. | |
| Mar 24, 2012 at 19:21 | history | asked | Samuel Reid | CC BY-SA 3.0 |