Provability, Computability and Reflection

absolute between QI assume axiom of choice card(x closed unbounded cofinal construction Corollary countable cumulative type structure define definition Dom(f element elementary embedding elementary substructure equivalent Exercise exp exp fact filter finite first-order fixed point follows formula free variables give given Gödel hence holds hypothesis II:-indescribable implies induction infinite initial ordinal initial segment isomorphism k-additive k-complete K(QI language large cardinals Lemma limit ordinal linear ordering Mahlo cardinals measurable cardinal model of ZF normal function normal measure Note notion order type parameters partition Proof prove QI F quantifiers result satisfies second-order sentences sequence set of indiscernibles set theory Skolem functions Souslin tree strongly compact cardinals strongly inaccessible cardinals subset Suppose TC(x transitive model transitive set weakly compact well-founded well-ordering write ZF H