## Infinitary combinatorics and the axiom of determinateness |

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array axiom of choice axiom of determinateness bk f cf(n Chapter clearly cofinal column combinatorial construction contains an u-closed contradiction countable choice countable ordinal countably additive definition of G equivalence class exponent partition relations fact function given partition homogeneous for F homogeneous set i-interlaced implies infinite descending chain infinite exponent partition initial segment integer Jonsson filter K-sequences large cardinal large cardinal properties least member least ordinal Lebesgue measurable let f let G limit ordinal Martin measurable cardinal member of Q nonconstant point normal measure notation numbers order-preserving order-type ordinal less partition F play proof of 4.3 Proof of claim prove Ramsey's theorem regular cardinal regular uncountable cardinal Rowbottom cardinal satisfies set of reals set theory sf H Solovay strategy for player Suppose f tells Turing degrees u-closed unbounded sets u-closed unbounded subset u-sequence ultrafilter ultrapower well-defined well-ordered choice winning strategy