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Let PD stand for projective determinacy, and consider the two claims:

(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)

(2) Con(ZFC+PD) implies Con(ZFC + there are infinitely many Woodin cardinals).

Clearly (2) implies (1).

As far as I know, (1) is true, and I have seen casual claims that (2) is true, but I am not sure such claims can be substantiated.

Can someone confirm these please? References to the literature would be of great help.

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It is also mentioned in mathoverflow.net/questions/73121/… So let us wait for Joel to give the reference. ;) –  The User Jun 20 '13 at 20:42

1 Answer 1

up vote 16 down vote accepted

(2) is false. (1) is true. In fact, $\mathsf{ZFC}+\mathsf{PD}$ (with $\mathsf{PD}$ stated as an axiom schema) implies that for every $n$ there is an inner model of $\mathsf{ZFC}$ with $n$ Woodin cardinals. Also, $\mathsf{ZFC}+\mathsf{PD}$ is equiconsistent with the theory that result from adding to $\mathsf{ZFC}$ the axiom schema, the $n$-th axiom of which states that there are $n$ Woodin cardinals.

This theory, in turn, is strictly weaker than $\mathsf{ZFC}+$"there are infinitely many Woodin cardinals", where of course the latter can be stated as $\forall n\,($there are $n$ Woodin cardinals$)$. This theory is equiconsistent with $\mathsf{ZFC}+L(\mathbb R)\models\mathsf{AD}$ or, if you prefer, with $\mathsf{ZF}+\mathsf{AD}$.

You can think of the difference this way: $\mathsf{PD}$ is only determinacy for sets of reals in $L_{\omega+1}(\mathbb R)$, which is far from determinacy for all sets of reals in $L(\mathbb R)$.

The paper

Peter Koellner, and W. Hugh Woodin. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, Matthew Foreman, and Akihiro Kanamori, eds, pp. 1951–2119, Springer, Dordrecht, 2010. MR2768702,

states the difference nicely, in Theorems 8.2 and 8.3:

8.2 Theorem. The following are equivalent: 1. $\mathsf{PD}$ (Schematic). 2. For every $n \lt \omega$, there is a fine-structural, countably iterable inner model $M$ such that $M\models$ There are $n$ Woodin cardinals.

8.3 Theorem. The following are equivalent: 1. $\mathsf{AD}^{L(\mathbb R)}$. 2. In $L(\mathbb R)$, for every set $S$ of ordinals, there is an inner model $M$ and an $\alpha<\omega_1$ such that $S\in M$ and $M \models \alpha$ is a Woodin cardinal.

This section of the paper is a survey, but proofs of this can be found in several places. I suggest

Ralf Schindler, and John Steel. The core model induction. Preprint, April 12, 2013,

though it has many prerequisites and is perhaps not the easiest source for these results, but it illustrates precisely the role of inner model theory at the level of Woodin cardinals in establishing these equivalences (or any consistency strength lower bounds that lie at the level of determinacy), and how both $\mathsf{PD}$ and $\mathsf{AD}^{L(\mathbb R)}$ are but natural stopping points of a much longer hierarchy.

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Super - many thanks! The third paragraph in your answer was especially illuminating (I was aware of most of the results above and the Koellner-Woodin article, but in a casual way). Your answer is important because on multiple occasions I have come across sloppy statements claiming equiconsistency (or even equivalence) of PD with the existence of infinitely many Woodin cardinals. –  Dave Albert Jun 20 '13 at 21:51
    
Glad I could help. –  Andres Caicedo Jun 20 '13 at 22:07

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