The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings

analogous argument Assume Axiom Baire property Borel sets characterization closed unbounded coding cofinality combinatorial concept consequences consistency strength contradiction corresponding countable crit(j defined descriptive set theory elementary embedding equivalent established existence extended filter finite follows formulated function Gödel Hence hierarchy homogeneous implies inaccessible cardinal indiscernibles induction infinite inner model theory iterable Jónsson K-complete Kechris Kunen large cardinals Lebesgue measurable Lemma limit ordinal Mahlo Martin mathematics measurable cardinal Moschovakis normal ultrafilter numbers ordertype partition property perfect set property prewellordering property proof Proposition recursive regular relation relative consistency result satisfying sequence set of reals Skolem term Solovay Solovay's stationary strong strongly compact structure subset suffices supercompact Suppose surjection Suslin Theorem transfinite tree Ult(V ultrapowers V₁ w₁ w₁-complete well-founded well-ordering winning strategy Woodin cardinals