Timeline for End extensions of models which do not preserve axioms
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2011 at 15:59 | comment | added | Emil Jeřábek | I see. Well, I find inconsistent large cardinals rather unpleasant, but whatever. Large cardinals are not the point, the point is that you can take as strong a theory as you want, as long as it does not prove things that are too obviously false. | |
| Nov 2, 2011 at 15:43 | comment | added | Joel David Hamkins | Emil, I took Andreas to mean that your comment suggests that all the large cardinals that please us are actually also consistent... | |
| Nov 2, 2011 at 14:13 | comment | added | Emil Jeřábek | (In case I was unclear, I’m referring to the arithmetical hierarchy, not Levy hierarchy.) Under normal circumstances, the $\Sigma^0_1$-soundness of a particular large cardinal is implied by the existence of a larger large cardinal (or even of two specimens of the same cardinal). If the large cardinals in whose consistency you believe can be put in an increasing chain, you are covered. | |
| Nov 2, 2011 at 12:59 | comment | added | Andreas Blass |
@Emil: The combination of "$\Sigma_1$-sound" near the beginning of your message and "whatever large cardinals you please" near the end suggests considerable confidence in my taste in large cardinal axioms. Thank you.
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| Nov 2, 2011 at 11:38 | comment | added | Emil Jeřábek | The actual McAloon theorem is much stronger. It says that for every $\Sigma_1$-sound recursively axiomatized theory $T\supseteq I\Sigma_1$, you can find (arbitrarily short) nonstandard initial segments that are models of $T$. For example, one can take for $T$ the set of all arithmetical consequences of ZFC (plus whatever large cardinals you please). Moreover, the theorem also holds for nonstandard models of the weaker theory $IE_1$ in place of $I\Delta_0$, as shown by Wilmers. | |
| Nov 2, 2011 at 9:34 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 135 characters in body
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| Nov 2, 2011 at 9:23 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |