Timeline for What axioms are used to prove Gödel's Incompleteness Theorems?
Current License: CC BY-SA 4.0
17 events
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| Feb 16 at 1:14 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Feb 16 at 0:01 | history | edited | LSpice | CC BY-SA 4.0 |
Links to comments, while this is on the front page
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| Jan 7, 2013 at 21:45 | history | edited | Ali Enayat | CC BY-SA 3.0 |
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| Jan 7, 2013 at 21:42 | comment | added | Ali Enayat | @Emil: thanks again; comedy of errors at work. | |
| Jan 7, 2013 at 19:00 | comment | added | Emil Jeřábek | Ali, the totality of $2^{2|x|}$ is not an equivalent formulation of $\Omega_1$, it is simply wrong. I think you are confused by the fact that the subscripts of $\omega_i$ in Hájek and Pudlák are off by one from both the usual convention and from the subscripts in $\Omega_i$. | |
| Jan 7, 2013 at 18:56 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix the definition
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| Jan 7, 2013 at 18:41 | history | edited | Ali Enayat | CC BY-SA 3.0 |
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| Jan 7, 2013 at 18:34 | comment | added | Ali Enayat | @Emil: thanks for your comments. I will add a note on the answer. | |
| Jan 7, 2013 at 14:26 | comment | added | Emil Jeřábek | And of course, the incompleteness theorem is provable in much weaker fragments than $I\Delta_0+\Omega_1$, such as $\mathit{PV}_1$. | |
| Jan 7, 2013 at 14:22 | comment | added | Emil Jeřábek | @Ali: Goldstern is right. Your definition is an inessential variant of $x^2$, and is provably total in $I\Delta_0$. | |
| Jan 6, 2013 at 16:23 | comment | added | Ali Enayat | @Martin: Thanks, I fixed the def. of $\Omega_1$. The right one is between what I had written and what you suggested; see p.272 of the Hájek-Pudlák text mentioned in my comment to Andrew (it is in Ch. V of the book). | |
| Jan 6, 2013 at 16:17 | history | edited | Ali Enayat | CC BY-SA 3.0 |
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| Jan 6, 2013 at 16:15 | comment | added | Ali Enayat | @Andrew: The second paragraph is not a quotation; the highlighting was used for journalistic reasons. The standard reference on the subject is the book "Metamathematics of First-Order Arithmetic" by Petr Hájek, and Pavel Pudlák (1998) [see the first chapter for $I\Delta_0 + exp$, and the last chapter for $I\Delta_0 + \Omega_1$]. This book is available on Project Euclid at projecteuclid.org/…. | |
| Jan 6, 2013 at 15:11 | comment | added | Andrew Critch | Thanks for this! Is that grey box a quotation? If so, where from? Where would you recommend I get started reading about this? | |
| Jan 6, 2013 at 15:05 | vote | accept | Andrew Critch | ||
| Jan 6, 2013 at 12:54 | comment | added | Goldstern | Really $2^{|x|}$, not $2^{|x|^2}$? | |
| Jan 6, 2013 at 8:06 | history | answered | Ali Enayat | CC BY-SA 3.0 |