Timeline for Failure of Shoenfield's Absoluteness
Current License: CC BY-SA 3.0
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| Nov 27, 2013 at 13:53 | comment | added | Ohad Drucker | This is obviously an excellent answer to my question, Thanks Andres ! Still, if anyone can think of a counterexample which is more natural, in the sense that it is in $ZF$ and have nothing to do with Shoenfield's absoluteness, I'll be glad if you could post it here. Just to clarify what I mean, failure of absoluteness for $\Sigma^1_3$ formulas is demonstrated with statements about constructible elements - definitely more natural, isn't it ? | |
| Nov 27, 2013 at 13:46 | comment | added | Ohad Drucker | Thanks ! So if I may rephrase it as a $ZF$ counterexample : Let $T \subseteq ZF$ be a finite fragment of $ZF$ that proves Shoenfield's Absoluteness. Choose $\kappa$ large enough so that $V_\kappa \models T$, and $M \subseteq V_\kappa$ transitive and of minimal height between models of $T$. Then $V_\kappa$ thinks that there is a well founded model of $T$, while $V_\kappa$ disagrees. | |
| Nov 27, 2013 at 13:42 | vote | accept | Ohad Drucker | ||
| Nov 26, 2013 at 19:01 | history | answered | Andrés E. Caicedo | CC BY-SA 3.0 |