Timeline for Breaking the circularity in the definition of N
Current License: CC BY-SA 2.5
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| Jun 14, 2010 at 23:04 | comment | added | Harald Hanche-Olsen | @Carl: Yes, the existence of nonstandard models are indeed a fact of life and there is no way out of that. I think you're right on in your first comment. Re your second comment, now you are talking about the meta level and the formalization of things like well-formed formulas and proof, right? Lacking Kunen's book I cannot be sure, but I did not get the impression that this sort of question is at issue here. | |
| Jun 14, 2010 at 22:53 | comment | added | Carl Mummert | @Harald: I would guess that "circularity" refers to the fact that the following three concepts are all mutually dependent: "natural number", "string of symbols", "proof". All three of these rely on the same notion of "finiteness", so in the end if you are doubtful about whether "finite" is well defined you cannot use any of the three concepts to clarify the other two. But this assumes that you are doubtful about what "finite" means. If we simply accept "finite" as an undefined term, and axiomatize the properties that "finite" things have, this starts to sound more like Kunen's proposal. | |
| Jun 14, 2010 at 22:48 | comment | added | Carl Mummert | Yes, that is the circularity. But the inductive definition doesn't really help, in first-order logic: there are still nonstandard models that satisfy all the same first-order induction axioms as the standard model. These nonstandard models think that all their nonstandard numbers "must be" obtained by rule 3. Maybe what Kunen means is that, once we have axiomatized enough of the true properties of the natural numbers, we can use those axioms to prove interesting theorems, and we don't have to worry too much about nonstandard models at that point. But I also don't have the book. | |
| Jun 14, 2010 at 22:22 | history | edited | Harald Hanche-Olsen | CC BY-SA 2.5 |
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| Jun 14, 2010 at 21:52 | history | answered | Harald Hanche-Olsen | CC BY-SA 2.5 |