## Metamathematics of Constructive Theories of Effective Operations |

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1945-realizability almost-modulus almost-negative argument assertion Assn(y associate axiom of choice axioms Church's thesis classical arithmetic clause completes the proof computation contradiction Corollary CT proves definition Doctor of Philosophy equivalent extensional effective operation extensionally equal f is total finite follows formalized formula fp-realized free variables GBdel Hence Heyting's higher types hypothesis integer interpretation Kleene's Kreisel kTzyk least Lemma Markov's principle MP Markov's rule modulus of continuity mr-HRO realized Mzy is defined partial recursive function PE(e Pr(A prenex primitive recursive function primitive recursive well-ordering provability predicate provably total indices q rp recursion theorem recursive parameters recursively inseparable rule corresponding rule with recursive s-m-n theorem schema sequence number extending Shoenfield soundness theorem suffices to show Suppose p rp Teun Texn total functions total recursive function TPT(y TPT2 transfinite induction undefined otherwise Val(y verify viii)n yAxy