Timeline for Interpretation of the Second Incompleteness Theorem
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2022 at 12:10 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
minor typos
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| May 28, 2022 at 14:08 | history | edited | Ira Gessel | CC BY-SA 4.0 |
Fixed ö in Gödel
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| Mar 3, 2016 at 21:56 | answer | added | abo | timeline score: -1 | |
| Feb 24, 2016 at 18:45 | answer | added | Nik Weaver | timeline score: 11 | |
| Feb 24, 2016 at 14:21 | history | edited | Stefan Geschke | CC BY-SA 3.0 |
added 1 character in body
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| Jan 6, 2013 at 12:17 | comment | added | Adam Epstein |
@Andres Can you elaborate about: But in $V_\kappa$ there are still plenty of (countable, transitive) models of ${\sf ZFC}+$there is an inaccessible''? Would this be via some Reflection Principle argument? Would it be fair to say that the proof of the stronger result $Con(ZFC)\rightarrow Con(ZFC+no inaccessible cardinals)$ does not similarly bypass the Second Incompleteness Theorem. Would these in fact be related considerations?
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| Aug 27, 2012 at 10:52 | comment | added | Rafał Gruszczyński | The point raised by Andreas and Kaveh below is nicely discussed in Craig Smorynski's "The Incompleteness Theorems", Handbook of Mathematical Logic (ed. by J. Barwise). Smorynski gives there a formal version of an argument for impossibility of Hilbert's Consitency Program in light of the 2nd Incompleteness Theorem. | |
| Aug 17, 2012 at 13:37 | history | edited | Stefan Geschke | CC BY-SA 3.0 |
edited body
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| Sep 10, 2010 at 10:58 | answer | added | nickname | timeline score: -4 | |
| Sep 9, 2010 at 20:18 | answer | added | Henry Towsner | timeline score: 7 | |
| Sep 9, 2010 at 20:00 | vote | accept | Stefan Geschke | ||
| Sep 9, 2010 at 19:59 | comment | added | Stefan Geschke | @Andres: I was thinking that observation (*) already invalidates Hilberts program to some extent (the consistency part). The First Incompleteness Theorem takes care of another issue: No reasonable system of axioms for mathematics is complete. But Andreas below has a real point. | |
| Sep 9, 2010 at 19:51 | comment | added | Stefan Geschke | @John and Andres: Thanks for the comments. I understand now that my example was poorly chosen, even though Andres' comment somehow rectifies this. What I wanted to say is that of course we can use the second incompleteness theorem to show that certain statements are unprovable in ZFC. This also addresses Peter Arndt's comment. Yes, I am aware of the usefulness of the Second Imcompleteness Theorem. What I am asking is "what are the philosophical implications of the theorem". | |
| Sep 9, 2010 at 19:20 | comment | added | John Stillwell | @Andres: It's a good point that Zermelo's incompleteness theorem is weaker the Goedel's. Still, I think it is interesting that unprovability of inaccessibles does not require the second incompleteness theorem, and that it was discovered earlier. | |
| Sep 9, 2010 at 17:59 | answer | added | Andreas Blass | timeline score: 34 | |
| Sep 9, 2010 at 17:37 | answer | added | user8248 | timeline score: 2 | |
| Sep 9, 2010 at 17:24 | answer | added | Carl Mummert | timeline score: 6 | |
| Sep 9, 2010 at 17:17 | comment | added | Peter Arndt | About (*): If we could prove the consistency of ZFC inside ZFC we would have shown the inconsistency of ZFC - and thus gained interesting information, isn't it? | |
| Sep 9, 2010 at 17:01 | answer | added | Kaveh | timeline score: 12 | |
| Sep 9, 2010 at 16:49 | comment | added | Andrés E. Caicedo | @Stefan: Do you think the first theorem already invalidates Hilbert's program? | |
| Sep 9, 2010 at 16:49 | comment | added | Andrés E. Caicedo | @John: But in $V_\kappa$ there are still plenty of (countable, transitive) models of ${\sf ZFC}+$``there is an inaccessible'', so this sense of incompleteness is certainly weaker. | |
| Sep 9, 2010 at 16:45 | comment | added | John Stillwell | In the case of inaccessible cardinals, you can bypass the second incompleteness theorem in the following sense. If an inaccessible exists, then there is a least inaccessible $\kappa$, and its existence is not provable because $V_\kappa$ is a model of ZF+"there is no inaccessible". (I like to call this Zermelo's incompleteness theorem, because he proposed the argument in 1928, before Goedel.) | |
| Sep 9, 2010 at 16:28 | history | asked | Stefan Geschke | CC BY-SA 2.5 |