Timeline for Breaking the circularity in the definition of N
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2013 at 11:53 | comment | added | Carl Mummert | @Kaveh: in most of the reverse mathematics literature, $\mathbb{N}$ is used for an arbitrary, possibly nonstandard model and $\omega$ is used for the standard model. But although this is common in that field, it is not universal. | |
| Jul 15, 2013 at 9:37 | comment | added | Kaveh | @Carl, my impression was that $\omega$ is used when the emphasis is on the set or at most an ordered set and $\mathbb{N}$ is used when one wants to put emphasis on the structure of natural numbers usually including at least addition and multiplication and $N$ is used for possibly nonstandard objects in models satisfying those properties of $\mathbb{N}$ that are expressible in the language. | |
| Jun 15, 2010 at 13:31 | comment | added | Carl Mummert | On notation: one convention in computability theory is to use ω to refer to the standard natural numbers and blackboard bold N to refer to an arbitrary model of (some fragment of) arithmetic. This is related to the use of the term "ω-model". But this convention is not universal; Kaye's book uses blackboard bold N for the standard model, and Kossak/Schmerl use both b.b. N and ω for the standard model. I have never seen a book that uses ω to refer to a nonstandard model, though. | |
| Jun 15, 2010 at 13:24 | comment | added | Marc Alcobé García | In other words, we avoid circularity at the expenses of precisely stating what we'd like to mean. | |
| Jun 15, 2010 at 13:18 | comment | added | Marc Alcobé García | As Tim mentioned in his answer to my other post, sometimes model theorists studying nonstandad models of arithmetic refer to N as the standard model, and to Th(N) as "true arithmetic", i. e. the set of sentences in the language of PA true in that model. Then they prove things like that if M is a nonstandard model of PA then M contains an isomorphic copy of N (in fact an initial segment of M). When one thinks informally about the natural numbers one has in mind N, not omega, which as Harald says is but a formal definition not capable of capturing what N really is. Hence the confusion. | |
| Jun 15, 2010 at 11:45 | vote | accept | Marc Alcobé García | ||
| Jun 15, 2010 at 11:22 | history | answered | Carl Mummert | CC BY-SA 2.5 |