Timeline for Does completeness of the theory of a bijection without finite orbits depend on choice?

8 events

when toggle format	what		by	license	comment
Jan 24 at 9:53	history	edited	Emil Jeřábek	CC BY-SA 4.0	added 63 characters in body
Jan 24 at 8:50	history	edited	Emil Jeřábek	CC BY-SA 4.0	more options
Jan 24 at 8:18	comment	added	Emil Jeřábek		Preservation of the truth of arithmetic sentences to transitive models of ZF such as $L$ follows immediately from the absoluteness of $\omega$. There is no need for anything as sophisticated as Shoenfield's absoluteness theorem.
Jan 23 at 21:29	comment	added	Timothy Chow		@GeorgeHayduke This is basically the Shoenfield absoluteness theorem.
Jan 23 at 16:29	comment	added	George Hayduke		I was hoping someone would come along and explain that part a bit. Where can I learn about this? It seems like black magic to me.
Jan 23 at 16:05	comment	added	Noah Schweber		Expanding on (3) (which applies to all theories) for the OP: the key point here is the definition and basic analysis of Godel's constructible universe $L$. For every finite $T\subset \mathsf{ZFC}$ we have a $\mathsf{ZF}$-proof that $T$ holds in $L$, and moreover we have a separate $\mathsf{ZF}$-proof that every arithmetical sentence true in $L$ is actually true; now given a $\mathsf{ZF}$-proof that your theory is complete, apply the above with $T$ the axioms used. None of this is quite trivial but once it's all put together we have a powerful black-boxable result for de-choiceifying arguments.
Jan 23 at 11:40	vote	accept	George Hayduke
Jan 23 at 11:22	history	answered	Emil Jeřábek	CC BY-SA 4.0