Theorems proved with AD whose proof is also known in the ZF world - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:09:42Z http://mathoverflow.net/feeds/question/79051 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79051/theorems-proved-with-ad-whose-proof-is-also-known-in-the-zf-world Theorems proved with AD whose proof is also known in the ZF world alephomega 2011-10-25T05:24:18Z 2011-10-29T06:47:10Z <p>This question arises from discussions with my professor and from Todd Eisworth comments in this question <a href="http://mathoverflow.net/questions/78863/large-cardinal-axioms-and-the-perfect-set-property" rel="nofollow">http://mathoverflow.net/questions/78863/large-cardinal-axioms-and-the-perfect-set-property</a></p> <p>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? </p> <p>So my question is: which theorems proved with $AD$ also have a known proof in the $ZF$+large cardinals world?</p> http://mathoverflow.net/questions/79051/theorems-proved-with-ad-whose-proof-is-also-known-in-the-zf-world/79092#79092 Answer by Russell May for Theorems proved with AD whose proof is also known in the ZF world Russell May 2011-10-25T15:55:54Z 2011-10-25T15:55:54Z <p>Perhaps, this is along the lines of what you're looking for. This <a href="http://digital.library.unt.edu/ark%3A/67531/metadc2789/m1/1/high_res_d/dissertation.pdf" rel="nofollow">thesis</a> 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 $&lt;\aleph_{\omega_2}$ in $L[\mathbb{R}]$ must collapse in $V$.</p> http://mathoverflow.net/questions/79051/theorems-proved-with-ad-whose-proof-is-also-known-in-the-zf-world/79452#79452 Answer by Trevor Wilson for Theorems proved with AD whose proof is also known in the ZF world Trevor Wilson 2011-10-29T06:47:10Z 2011-10-29T06:47:10Z <p>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.</p>