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This question arises from discussions with my professor and from Todd Eisworth comments in this question Large cardinal axioms and the perfect set property

In $L(\mathbb{R})$ we have $AD$ and it is a powerful tool to prove theorems. Almost all of the theorem proved with $AD$ come in a very natural way: we use games and determinacy as in the First/Second/Third Periodicity Theorems. However no "$ZF$+Large Cardinal" proof is known for the Periodicity Theorems. Another example is that of the Perfect Set Property: Using $AD$ all sets of reals have the Perfect Set Property, but is a proof of the statement "Assuming infinitely many Woodin cardinals with a measurable above then every set of reals has the perfect set property" known?

So my question is: which theorems proved with $AD$ also have a known proof in the $ZF$+large cardinals world?

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  • $\begingroup$ Of course, AD has large cardinal consistency consequences, which are also directly provable in ZFC + large cardinals. $\endgroup$ Oct 25, 2011 at 12:38
  • $\begingroup$ You probably need to sharpen this question up a bit. For example, ZFC implies that there is a set of reals without the perfect set property, so of course we can't prove (in ZFC+large cardinals) that every set of reals has the perfect set property. Do you mean to ask about ZF? $\endgroup$ Oct 25, 2011 at 14:32
  • $\begingroup$ @Todd, Thx for the correction. Of course I can't say ZFC. $\endgroup$ Oct 25, 2011 at 22:12

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Perhaps, this is along the lines of what you're looking for. This thesis gives a proof of the stong partition relation on $\omega_1$ from AD, and then "relativizes" the proof to $V$ to show, assuming the existence of Woodin cardinals, a collapsing result, namely, that some regular cardinal $<\aleph_{\omega_2}$ in $L[\mathbb{R}]$ must collapse in $V$.

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    $\begingroup$ Perhaps as a full disclosure you should add that the thesis is actually your thesis. $\endgroup$
    – Asaf Karagila
    Oct 25, 2011 at 17:51
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    $\begingroup$ @Asaf Kargila: I thought about doing that, but I think it's fairer to say that all the major ideas there were from my advisor, Steve Jackson. $\endgroup$ Oct 25, 2011 at 18:07
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I don't know if this is what you are looking for, but many important examples of statements that are consequences both of AD and of large cardinals are themselves phrased in terms of determinacy or large cardinals. For example, AD and "there is a measurable cardinal" both imply "every analytic set is determined" and "every real has a sharp". (In fact, these two conclusions are equivalent to one another.) There are more such phenomena at other levels of consistency strength.

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