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.



Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$
The only reference I know for precisely these matters is the handbook chapter 



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> [MR2768702](https://mathscinet.ams.org/mathscinet-getitem?mr=2768702). Koellner, Peter; Woodin, W. Hugh. *[Large cardinals from determinacy](https://link.springer.com/chapter/10.1007%2F978-1-4020-5764-9_24)*. In **Handbook of set theory. Vols. 1, 2, 3**, 1951–2119, Springer, Dordrecht, 2010. 
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(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.


