Talk:Independence from ZFC: Difference between revisions
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The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. [[User:XnoobSpeakable|XnoobSpeakable]] | The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. [[User:XnoobSpeakable|XnoobSpeakable]] | ||
: Currently, all of the machines listed in this article except one are written to start enumerating theorems of ZF-Regularity and halt if they find a contradiction, so they halt iff Con(ZF-Regularity) is false. Con(ZF-Regularity) is equivalent to Con(ZF) although I don't know how to prove that, but I do know how to prove that Con(ZF) is equivalent to Con(ZFC): if you assume ZF is consistent, there is a model of it, then you take the constructible universe of that model to obtain a model of ZFC, so then ZFC is consistent. The one different machine is the original Aaronson-Yedidia machine. Instead it uses one of Friedman's statements, which is independent of both ZF and ZFC (and even ZFC+some large cardinals!) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:04, 21 July 2025 (UTC) | |||
Revision as of 20:04, 21 July 2025
A good introduction can come from here: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf,
ZF vs. ZFC
The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. XnoobSpeakable
- Currently, all of the machines listed in this article except one are written to start enumerating theorems of ZF-Regularity and halt if they find a contradiction, so they halt iff Con(ZF-Regularity) is false. Con(ZF-Regularity) is equivalent to Con(ZF) although I don't know how to prove that, but I do know how to prove that Con(ZF) is equivalent to Con(ZFC): if you assume ZF is consistent, there is a model of it, then you take the constructible universe of that model to obtain a model of ZFC, so then ZFC is consistent. The one different machine is the original Aaronson-Yedidia machine. Instead it uses one of Friedman's statements, which is independent of both ZF and ZFC (and even ZFC+some large cardinals!) C7X (talk) 20:04, 21 July 2025 (UTC)