Martin's Axiom and Determinacy-axioms: independence results - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:35:20Z http://mathoverflow.net/feeds/question/63821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63821/martins-axiom-and-determinacy-axioms-independence-results Martin's Axiom and Determinacy-axioms: independence results Matteo Mio 2011-05-03T15:32:35Z 2011-05-03T15:58:42Z <p>Hello everybody,</p> <p>I'm searching for references for the following independence assertions:</p> <p>ZFC + $MA_{\aleph_{1}}$ $\not\vdash$ "Analytic determinacy"</p> <p>ZFC + $MA_{\aleph_{1}}$ $\not\vdash$ $\neg$ ("Analytic determinacy")</p> <p>i.e. $MA_{\aleph_{1}}$ does not settle any determinacy question. The question extends also to Projective determinacy.</p> <p>Also I'd need references for the reversed independence question, i.e. Analytic determinacy (and Projective det. ) does not settle cardinality issues, so for instance.</p> <p>ZFC + Analitic-Determinacy $\not\vdash$ CH</p> <p>ZFC + Analitic-Determinacy $\not\vdash$ $\neg$CH</p> <p>but also</p> <p>ZFC + Analitic-Determinacy + "$2^{\aleph_{0}}> \aleph_{1}$" $\not\vdash$ $MA_{\aleph_{1}}$</p> <p>and</p> <p>ZFC + Analitic-Determinacy + "$2^{\aleph_{0}}> \aleph_{1}$" $\not\vdash$ $\neg MA_{\aleph_{1}}$</p> <p>where by $MA_{\aleph_{1}}$ I mean the standard instance of Martin's Axiom at $\aleph_{1}$ (which implies $\neg CH$). </p> <p>Please note that I have at my hands Fremlin's book "Consequences of Martin's Axiom" but it is <strong>very</strong> hard to read, and in the summary I couldn't find even the work "analytical determinacy" and just a reference to "determinacy". I also have Jech's Set theory. However I need these references for my PhD thesis (just to mention these facts) which i'm writing right now, and I'd rather not invest too much time searching in books at this stage. So please, if you can, provide precise references.</p> <p>THank you very much,</p> <p>bye</p> <p>matteo</p> http://mathoverflow.net/questions/63821/martins-axiom-and-determinacy-axioms-independence-results/63825#63825 Answer by Andreas Blass for Martin's Axiom and Determinacy-axioms: independence results Andreas Blass 2011-05-03T15:58:42Z 2011-05-03T15:58:42Z <p>ZFC plus <code>$\text{MA}_{\aleph_1}$</code> is consistent relative to ZFC, while analytic determinacy has a little bit of large cardinal strength, namely the existence of sharps of reals. So ZFC+MA cannot prove analytic determinacy. On the other hand, analytic determinacy follows from the existence of a measurable cardinal, and the usual way of forcing <code>$\text{MA}_{\aleph_1}$</code> preserves measurable cardinals (being a small forcing). So ZFC+<code>$\text{MA}_{\aleph_1}$</code> can't refute analytic determinacy either. Similarly, since you can force either of CH or not-CH with a small forcing, hence preserving measurable cardinals, analytic determinacy cannot decide CH. </p>