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A-primitive A-recursively A-semicomputable algorithm alphabet axiom called Chap characteristic function combinatorial system completely computable functional completely recursive computable functional consist constructive Corollary decision problem defined Definition 1.5 diophantine predicate domain equations exists finite number first-order logic follows at once given Godel number Hence hq'h Immediate from Theorems instantaneous description integers internal configuration Kleene hierarchy Lemma mathematical modus ponens n-tuples natural numbers normal system numerical predicate obtained ordinal P(xi partial propositional calculus partial recursive function partially A-computable function partially computable prime primitive recursive function problem is recursively productions propositional calculus proved qi Sj qL+i qp+u qw+i recursive set recursively enumerable set recursively solvable respect rm(Z semi-Thue system semicomputable predicate semigroup simple Turing machine singulary suppose Theorem 3.1 theory Thue system total function true undefined unsolvable decision problem variable word write