Timeline for Did Edward Nelson accept the incompleteness theorems?
Current License: CC BY-SA 4.0
14 events
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| Dec 21, 2020 at 15:58 | comment | added | Timothy Chow | @DavidRoberts : The result perhaps says more about the concept of interpretability than anything else. Solovay's interpretation is not a cut-interpretation. The domain formula is not downward closed with respect to $\le$ and hence you have a lot of room in which to play tricks. | |
| Dec 21, 2020 at 10:24 | comment | added | David Roberts♦ | @TimothyChow ok, thanks! It's a really counterintuitive result, from my non-expert viewpoint. | |
| Dec 20, 2020 at 15:24 | comment | added | Timothy Chow | @DavidRoberts : Ferreira and Ferreira prove the result in Section 8 of their paper and they attribute it to Solovay. "We are grateful to Robert Solovay for the kind permission to report on his old unpublished results obtained in the mid nineteen eighties (see Section 8)." | |
| Dec 20, 2020 at 10:47 | comment | added | David Roberts♦ | (the latter is a result of Solovay) <-- citation? | |
| Dec 20, 2020 at 1:12 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added link to paper by Wilkie and Paris
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| Dec 15, 2020 at 18:57 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Major rewrite
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| Dec 13, 2020 at 16:47 | comment | added | Emil Jeřábek | Yes, though I guess the $\Sigma^b_1$ version can be made into a single statement using the existence of a universal $\Sigma^b_1$ formula in a fragment of $I\Delta_0+\Omega_1$. | |
| Dec 13, 2020 at 16:33 | comment | added | Timothy Chow | @EmilJeřábek : Ah, so it's really "Goedel's 2nd incompleteness theorem schema" so to speak? | |
| Dec 13, 2020 at 15:24 | comment | added | Emil Jeřábek | The statement doesn’t really quantify over the theories $T$. You are given a $\Sigma_1$-formula (or better, $\Sigma^b_1$ formula, to avoid additional difficulties) that defines an axiom set for $T$. | |
| Dec 13, 2020 at 14:07 | comment | added | Timothy Chow | @EmilJeřábek : I understand that syntax can be arithmetized in weak theories but how exactly are you stating "T is recursively axiomatized theory" in a weak system? Also, as I explained, although Nelson surely accepts that all the reasoning in his book is valid, it does not follow that he believes the theorem statements in the sense that he regards the theorems as making meaningful assertions that he actually believes. | |
| Dec 13, 2020 at 10:49 | comment | added | Emil Jeřábek | After a closer look at the book, it turns out that arithmetizing syntax, including consistency statement, in $I\Delta_0+\Omega_1$, is exactly what Nelson is doing in the second part of the book! So, assuming we can take at face value that he accepts as valid everything that he put in the book, there is no doubt that he finds arithmetization of consistency statements (including the formulation of Gödel’s second incompleteness theorem) as meaningful, and he even developed himself in the book more or less all the tools needed to actually prove the incompleteness theorem. | |
| Dec 13, 2020 at 8:27 | comment | added | Emil Jeřábek | ... to actually interpret, say, $I\Delta_0+\Omega_1+Con(T)$ by passing to a definable cut if necessary, there is no question whether the statement really "means" the consistency of $T$ (or rather, this is no more in question than it is when applied in PA). Sequence encoding and formalization of the various syntactic notions used in the consistency statements are perfectly well-behaved in $I\Delta_0+\Omega_1$. | |
| Dec 13, 2020 at 8:20 | comment | added | Emil Jeřábek | The Bezboruah and Shepherdson result (which basically exploits the fact that Q doesn't prove much of anything to show that in particular, it doesn't prove anything of a syntactic form similar to a consistency statement) is a red herring. The real second incompleteness theorem for weak theories says that if $T$ is any consistent recursively axiomatized theory, then $T$ cannot interpret $Q+Con(T)$. (Pudlák, Cuts, consistency statements, and interpretations, JSL 1985, proves an even stronger form with restricted consistency statements.) Since an interpretation of $Q+Con(T)$ can be assumed ... | |
| Dec 13, 2020 at 6:08 | history | answered | Timothy Chow | CC BY-SA 4.0 |