Determinacy and Choice

Question

I know that Kechris proved $$\text{Con}(\text{ZF} + \text{AD})\Rightarrow \text{Con}(\text{ZF} + \text{AD} + \text{DC})$$ And that $\text{DC}$ and $\text{AC}_\omega$ are independent from $\text{AD}$, by results of Solovay and Woodin.

Can we say $$\text{AD} \Rightarrow \text{DC}(\omega^\omega)?$$

what about $$\text{Con}(\text{ZF} + \text{PD})\Rightarrow \text{Con}(\text{ZF} + \text{PD} + \text{AC})?$$

If the last consistency relation does not hold, what can we say about the relationship between projective determinacy and choice?

Thanks!

Answer 1

It appears that whether $\mathsf{AD}$ implies $\mathsf{DC}_\mathbb{R}$ is open. I believe there is a comment in Larson's $\mathsf{AD}^+$ book which states that Woodin has shown that $\mathsf{Con(ZF + AD + \neg DC_\mathbb{R})}$ implies $\mathsf{Con(ZF + AD_\mathbb{R})}$. It is known that $\mathsf{AD}_\mathbb{R}$ has stronger consistency strength than $\mathsf{AD}$. (Of course, having stronger consistency strength here could be the result of being inconsistent.) This question is related to whether $\mathsf{AD}$ and $\mathsf{AD}^+$ are equivalent.

Moschovakis proved that $\mathsf{Con(ZF + DC + PD)} \Rightarrow \mathsf{Con(ZFC + PD)}$. (See "Partially Playful Universes" Section 5 and Kechris and Moschovakis "Notes on Scales" Section 5. Also "Notes on Scales" Lemma 5.1 has a comment stating that dependent choice is not needed. )

Moschovakis proved this with a model called $M^\omega$ (whichi is related to the partially playful $M^n$ and not to be confused with the $M_n$ which are the inner models of Woodin cardinals). For each $n$, let $P_n \subseteq \omega \times \mathbb{R} \times\mathbb{R}$ be a lightface $\Pi_{n - 1}^1$ set universal for $\Pi_{n - 1}^1$ sets. Assuming projective determinacy (in particular the definable uniformization results), there is a $\Pi_n^1$ uniformization function $F_n$.

The model $M^\omega$ is defined by a $L$-like construction.

$M^\omega_0 = \emptyset$.

$M^\omega_{\alpha + 1} = M^\omega_\alpha \cup D(M^\omega_\alpha) \cup \{F_n(n,x) : n \in \omega \wedge x \in \mathbb{R} \cap M_\alpha^\omega\}$, where $D(M^\omega_\alpha)$ are all the definable subsets of $M^\omega_\alpha$.

If $\lambda$ is limit, then $M^\omega_{\lambda} = \bigcup_{\alpha < \lambda} M^\omega_\alpha$.

Let $M^\omega$ be the union of $M_\alpha^\omega$ over all ordinals $\alpha$. Basically $M^\omega$ is the result of closing off the models under the usual constructibility ideas as well as all the uniformization functions $F_n$. This is a model of $\mathsf{AC}$ and even $\mathsf{GCH}$ in much the same way as $L$. It can be shown by absoluteness arguments to be a model projective determinacy. See the references above for the full details. Using just one of the $F_n$ function, defines the partially playful models $M^n$ which only satisfy a limited amount of projective determinacy.