# Forcing absoluteness in the setting of second-order arithmetic

There are some results about (set-size) forcing absoluteness for first-order properties in $$L(\mathbb{R})$$ and for $$\mathbf{\Pi}^1_{\infty}$$ properties when one works over $$\mathsf{ZFC}$$. My question is how much of $$\mathbf{\Pi}^1_{\infty}$$ forcing absoluteness could be stated just in the language of second-order arithmetic and what is the exact relation between this and projective determinacy? Unfortunately I have fairly limited knowledge about both the forcing absoluteness and formalization of forcing in second-order arithmetic.

I assume that the base system of SOA is reasonably strong, it seems that $$\mathsf{Z}_2+\mathsf{DC}$$ should be a good choice here. It seems that in general there is an issue with formalizing forcing in SOA namely that names (for sets of naturals) are by necessity countable. This shouldn't be a problem when a poset is countable, i.e. encoded by a set. Also it seems not to be a problem for $$\mathbf{\Pi}^1_{\infty}$$-definable posets, where all antichains are at most countable (note that this condition by itself isn't formalizable in SOA and thus should be replaced with some stronger condition that is formalizable in SOA). But I don't know whether it works for some other classes of forcings. Thus a part of the question is what is the reasonable choice of the class of forcings to work with in the context of SOA. Given such class of forcings we then could naturally state $$\mathbf{\Pi}^1_{\infty}$$ forcing absoluteness as a scheme in the language of SOA.

It is possible to derive the upper bound for the consistency strength of this kind of absoluteness from the paper [1] to be $$\mathsf{ZFC}+\{\textrm{there is n strong cardinals}\mid n\in\omega\}$$. But how much weaker it will become if we restrict the forcing notions to that that are expressible in the language of SOA?

[1] Hauser, K. (1995). The consistency strength of projective absoluteness. Annals of pure and applied logic, 74(3), 245-295.

• If you restrict to forcings with countable antichains, then this shouldn't be very strong (however you formalize it) since an unpublished theorem of Kunen shows that even $L(\mathbb R)$ generic absoluteness for ccc forcings is equiconsistent with a weakly compact cardinal. This is mentioned in Schindler's "Proper forcing and remarkable cardinals," which is also relevant. – Gabe Goldberg Jul 22 '19 at 17:27
• @GabeGoldberg Thank you for the comment! So at least the more straightforward version of forcing absoluteness for second-order arithmetic will be relatively weak. – Fedor Pakhomov Jul 22 '19 at 18:35
• Actually, more precise (and published) results are mentioned in Noah's answer here: mathoverflow.net/questions/209000/… – Gabe Goldberg Jul 22 '19 at 18:42