# Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$

Let $Z_2$ and $Z_3$ be second and third order arithmetics respectively.

In $Z_2$'s language, $\text{AD}$ (the axiom of determinacy) and $\text{PD}$ (projective determinacy) are stated the same way (since projective sets are precisely the ones defined by formulas in $Z_2$'s language). But in $Z_3$'s language we can state $\text{AD}$ and $\text{PD}$ separately.

$Z_2+\text{PD}$, $Z_3+\text{PD}$ and in fact $\text{ZFC+PD}$ are all equiconsistent, and weaker than $\text{ZF+AD / ZFC+AD}^{L(\mathbb{R})}$. I was wondering if $Z_3+\text{AD}$ was also equiconsistent with those, or if it has the full consistency strength of $\text{ZF+AD / ZFC+AD}^{L(\mathbb{R})}$ (or falls somewhere inbetween). I would appreciate a reference or a proof sketch.

• How is $Z_3+\mathrm{PD}$ equiconsistent with $\mathrm{ZFC}+\mathrm{PD}$? The latter proves the consistency of the former. – Andrés E. Caicedo May 9 at 20:30
• I think there's a confusion here between PD, which is a single sentence, and the infinite set of sentences $\{\text{Det}(\mathbold{\Sigma}^1_n):n=1,2,\dots\}$. I could believe equiconsistency for the infinite sets; $Z_3$ plus a particular level of determinacy should give a model of ZFC plus a slightly lower level of determinacy. But for the single sentences, you definitely don't get equiconsistency. – Andreas Blass May 9 at 20:47
• An example of a true result along these lines is that $Z_2+\Delta^1_2$-determinacy is equiconsistent with $\mathrm{ZFC}+$"there is a Woodin cardinal". As Andreas suggests, you probably want to clarify whether you actually mean the schematic version of $\mathrm{PD}$. – Andrés E. Caicedo May 9 at 20:57
• @Andrés If ZFC holds and there is a Woodin cardinal $\delta$, doesn't $\Delta^1_2$ determinacy hold in $V[G]$ where $G \subset \text{Col}(\omega,\delta)$ is $V$-generic, giving a set model $(V_\omega, V_{\omega+1}; \mathord{\in})^{V[G]}$ of $Z_2 + \Delta^1_2$ determinacy? – Trevor Wilson May 10 at 2:18
• @Trevor Agh, sorry, that was a horrible typo. I meant to write "$\mathrm{ORD}$ is Woodin". But now I am not even sure this is quite correct; it seems to be open. In the Koellner-Woodin chapter in the Handbook, they state without proof that $Z_3+\Delta^1_2$-determinacy is equiconsistent with $\mathrm{ZFC}_2+$"$\mathrm{ORD}$ is Woodin", where $\mathrm{ZFC}_2$ is second-order $\mathrm{ZFC}$. Actually, that chapter is a good reference for these matters. For instance, they obtain that $Z_2+$ boldface $\Delta^1_2$-determinacy implies the consistency of $\mathrm{ZFC}+$"there is a Woodin cardinal", – Andrés E. Caicedo May 10 at 14:23

The only reference I know for precisely these matters is the handbook chapter

MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010.

(Particularly, section 7.)

For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harrington's principle.

First of all, $\mathsf{ZFC}+\mathsf{PD}$ proves the consistency of $Z_3+\mathsf{PD}$, and similarly $\mathsf{ZF}+\mathsf{AD}$ and $\mathsf{ZFC}+\mathsf{AD}^{L(\mathbb R)}$ prove the consistency of $Z_3+\mathsf{AD}$: (depending on your precise formulation of second- and third-order arithmetic), $(V_\omega,V_{\omega+1},V_{\omega+2},\in)^{L(\mathbb R)}$ is a model of the latter.

That said, however, (as pointed out by Andreas in a comment) there is a subtlety here in that $\mathsf{PD}$ is a single sentence. If instead we consider the infinite collection of axioms, the $n$th of which states the determinacy of boldface $\Sigma^1_n$-games, we should probably get equiconsistency, perhaps after some minor tweaking of the theories.

Section 7 of the reference above addresses these matters in reasonable detail for $\mathsf{ZFC}+$"there is a Woodin cardinal" and close variants, and $Z_2+$ lightface $\Delta^1_2$-determinacy, and close variants. For instance, from $Z_2+$ boldface $\Delta^1_2$-determinacy, they obtain (something stronger than) the consistency of $\mathsf{ZFC}+$"There is a Woodin cardinal". Note that the latter theory is equiconsistent with $\mathsf{ZFC}+$ lightface $\Delta^1_2$-determinacy which, in turn, implies the consistency of $Z_2+$ lightface $\Delta^1_2$-determinacy, so one needs to be careful here. They conjecture that $Z_2+$ lightface $\Delta^1_2$-determinacy is equiconsistent with $\mathsf{ZFC}+$"$\mathrm{ORD}$ is Woodin". In any case, from these results and the techniques in that chapter (particularly in section 6) one should obtain the equiconsistency results I suggested in the previous paragraph.