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abstraction term acceptable triple action of TT adding ordinals axiom of choice belong bels Chapter cofinalities constants are added contains labels contradict countable model countable sets Dedekind finite define a function define the action domain and range DOUGLASS BERT MORRIS efficient label elements fixed elements of E(c elements of rank elements of X1 f-condition Finite Order finitely many elements fix ordinals fixes all elements forcing argument forcing conditions ground model hold induction integers large cardinal Lemma lh(a limit ordinal metalanguage model N Model of ZF model with automorphisms n-tuples nice Cohen extension P-function partial automorphism permutation power set quasicondition ramified language ranked formulas replacement Section set of indiscernibles set theory sets of total statements subscripts supremum symmetry argument thesis tion ultrafilter University of Wisconsin unranked weakly forced well-ordered ZF Not Extendable ZF with Automorphisms ZF(J ZFC without adding ZFC(J