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Hello everybody,

I'm searching for references for the following independence assertions:

ZFC + $MA_{\aleph_{1}}$ $\not\vdash$ "Analytic determinacy"

ZFC + $MA_{\aleph_{1}}$ $\not\vdash$ $\neg$ ("Analytic determinacy")

i.e. $MA_{\aleph_{1}}$ does not settle any determinacy question. The question extends also to Projective determinacy.

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.

ZFC + Analitic-Determinacy $\not\vdash$ CH

ZFC + Analitic-Determinacy $\not\vdash$ $\neg$CH

but also

ZFC + Analitic-Determinacy + "$2^{\aleph_{0}}> \aleph_{1}$" $\not\vdash$ $MA_{\aleph_{1}}$

and

ZFC + Analitic-Determinacy + "$2^{\aleph_{0}}> \aleph_{1}$" $\not\vdash$ $\neg MA_{\aleph_{1}}$

where by $MA_{\aleph_{1}}$ I mean the standard instance of Martin's Axiom at $\aleph_{1}$ (which implies $\neg CH$).

Please note that I have at my hands Fremlin's book "Consequences of Martin's Axiom" but it is very 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.

THank you very much,

bye

matteo

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1 Answer 1

up vote 13 down vote accepted

ZFC plus $\text{MA}_{\aleph_1}$ 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 $\text{MA}_{\aleph_1}$ preserves measurable cardinals (being a small forcing). So ZFC+$\text{MA}_{\aleph_1}$ 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.

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Thanks Andreas for the good answer. It contains very relevant informations. Are you aware of any textbook or (survey level, i guess) paper which explicitly states the result and goes into some detail? I'm my thesis i use both $MA_{\aleph_{1}}$ and (some) determinacy axioms (below or equal to PD), but I just use their consequences and don't go (even in the introduction) in such things like forcing, small forcing, sharps or large cardinals. So it would be a bit difficult to give the desired result without actually have to design an entire new section in the introduction. thanks again! –  Matteo Mio May 3 '11 at 16:15
    
I would expect that everything I cited is in Jech's "Set Theory" (but I'm away from home so I can't just check). You probably won't need a whole new section to cover this; something like what I wrote, decorated with a few pointers to Jech should do the job. –  Andreas Blass May 3 '11 at 16:42
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10.1K ! Superpowers! Congratulations. :-) –  Andres Caicedo May 3 '11 at 16:52
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Let me add that there are well known strengthenings of MA that imply analytic determinacy and much more; for example, PFA (the proper forcing axiom). On the other hand, no amount of determinacy or large cardinals suffices to decide CH or to imply MA. –  Andres Caicedo May 3 '11 at 16:56
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@Andres: Thanks. So far, the only effect of superpowers was that MO showed me an answer that had been deleted by its author (because it was wrong). It's like having X-ray vision to look into people's trash cans. –  Andreas Blass May 4 '11 at 13:33
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