I have a question about projective determinancy that I was hoping someone could answer. Is projective determinancy provably consistent with ZFC , or does its consistency require large cardinals to prove?
Ars Mathematica
Dedicated to the mathematical arts.
i don’t understand the question (nor logic neccesarily).
according to wikip, PD is undecidable in ZFC. thats your answer, i think. you can then use large cardinals (assuming you believe in them) outside of ZFC to prove or disprove it (and there may be a more interesting question).
The more complete answer is that Projective Determinacy is provably equiconsistent with ZFC plus the existence of countably many Woodin cardinals. Borel Determinacy is provably from ZFC (though it makes essential use of the Axiom Schema of Replacement). I believe that n+2 Woodin cardinals are needed to prove determinacy for the first n levels of the projective hierarchy, though I may have gotten the “+2″ wrong.
It’s an an axiom, so it’s independent of ZFC.
Not all axioms are independent – in particular, the axiom of pairing is provable from powerset and replacement, and in some formulations, separation is provable from replacement as well. Some formulations also include the existence of the empty set as an axiom, but that is also provable from infinity and separation.
And I meant “Borel Determinacy is provable from ZFC”, rather than “provably”, which didn’t make any sense.
beats me.
the new _notices_ is out.
http://www.ams.org/notices/200703/200703-toc.html