Xref: utzoo sci.math:11873 comp.theory:914
Path: utzoo!attcan!uunet!cs.utexas.edu!samsung!sdd.hp.com!ucsd!ogicse!milton!uw-beaver!sumax!spector
From: spector@sumax.UUCP (Mitchell Spector)
Newsgroups: sci.math,comp.theory
Subject: Re: Infinite games
Summary: No large cardinal assumption is needed to prove Borel determinacy.
Keywords: Determinacy, Borel games.
Message-ID: <1567@sumax.UUCP>
Date: 2 Aug 90 20:42:58 GMT
References: <1033@fornax.UUCP>
Reply-To: spector%sumax.UUCP@beaver.cs.washington.edu (Mitchell Spector)
Organization: Seattle University, Seattle, WA
Lines: 22

In article <1033@fornax.UUCP> mahajan@fornax.UUCP (Sanjeev Mahajan) writes:
>It is known (I forget the reference) that all Borel games are determinate,
>(that is either their is a winning strategy for player I or there is
>a winning strategu for the player II) if measurable cardinals exist.

   Martin proved without any large cardinal assumption that all Borel
games are determined.  The proof does require a substantial amount of
ZF and cannot be carried out, for example, by discussing just reals and
sets of reals (as H. Friedman showed, actually before Martin's proof).

   The existence of a measurable cardinal implies that all analytic games
are determined; in fact, by work of Martin and Harrington, the determinacy
of analytic games is equivalent to the existence of r# for all reals r
(which follows from the existence of a measurable cardinal).

>Sanjeev 

--
Mitchell Spector
Dept. of Computer Science and Software Engineering
Seattle University
E-mail: spector%sumax.uucp@beaver.cs.washington.edu