Conservation of Hyperarithmetic Sentences over AC and CH.

Question

I know that arithmetic sentences are conserved under the addition of the axiom of choice and the continuum hypothesis to ZF (i.e. ($ZF+AC \vdash \phi$ iff $ZF \vdash \phi$) and ($ZF+CH \vdash \phi$ iff $ZF \vdash \phi$) for arithmetic sentences $\phi$) Does this result extend to analytic sentences or other hyperarithmetic sentences? If so, how far does it extend up the hyperarithmetic hierarchy?

Answer 1

First, let me remark that in your question, you can combine AC and CH together, rather than having two separate conservation results as you did. In fact, you can ramp CH up to GCH and more, including such principles as $\Diamond$ or others, without any problem. That is, the conservation result is that ZFC + GCH proves $\varphi$ if and only if ZF proves $\varphi$, for a large class of statements $\varphi$, including the arithmetic statements, as you mentioned, but much more.

The phenomenon extends completely up the hyperarithmetic hierarchy and beyond, beyond even the analytic sentences up into the projective hiearchary at the level of $\Sigma^1_2$. (In this hiearchy, the hyperarithmetic statements are $\Delta^1_1$ and the analytic statements $\Sigma^1_1$.)

This absoluteness result is the content of the Shoenfield Absoluteness Theorem, which asserts that any $\Sigma^1_2$ statement is absolute between between any two transitive models of set theory $V\subset W$ having the same ordinals. In particular, a $\Sigma^1_2$ statement holds in the universe if and only if it holds in the constructible universe $L$, where both AC and GCH hold.

Thus, the $\Sigma^1_2$ statements provable in ZFC+GCH are the same as those provable in ZF.