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abstract algebraically closed fields arithmetic automorphism axiomatic Borel cartesian closed category class of Woodin closure computable functionals Computer Science consistency construction continuous functionals Continuum Hypothesis countable definition denote editors elements equivalent Ershov example exists a proper explicit mathematics extensional Feferman finite type formula geometry hence hereditarily hierarchy higher type computability Hilbert Hrushovski ideals independence inductive infinite inner model isomorphism iteration Journal of Symbolic Kleene computable Kleene's language large cardinal axiom Lecture Notes Lemma Logic Colloquium Mahlo Martin's Maximum Mathematical Society metapredicative model theory Moschovakis natural numbers Normann North-Holland notions of computability operations ordinal paradoxes partial functions problem Projective Determinacy projective sets proof proof-theoretic proper class quotients recursion theory recursive functions Russell Section semantics sequence sequential set theory stationary subset strongly minimal Suppose Symbolic Logic Theorem theory of types transfinite Turing type structure universally Baire sets Woodin cardinals