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3 The relative hierarchy
5 The satisfaction predicates
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admissible axiom of choice axiom of dependent axiom of extensionality axiom of foundation axiom of infinity axiom schema basic language binary relation Boolean closure complete consistent extension consistent recursively axiomatizable constant terms contradicting Corollary 6(b defined denote dependent choices easily seen equivalent in Q essentially extended language extension of ZF Feferman first-order predicate calculus formulas of rank free variables Fv(m Godel Godel-number hence Q HIERARCHY THEOREM immune induction hypothesis Lemma 34 Lemma 7(a Lemma 9(a,b Let f Let Q natural number negation number theory obviously PR-formula proof in Q proof of Lemma proof of Theorem provable prove Q is productive quantifiers recursive set recursively axiomatizable extension recursively enumerable Rel(L Rel(t Rel(z restricted formula sentences of Q sentential connectives set of sentences set theory Stsf subset Tc(x theo Theorem 18 Theorem 23 Theorem 44 theory Q true sentence truth definition well-founded relation ZF+AC