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abstract algebra analysis applications arithmetic Association for Symbolic automorphism axiomatic Boolean Brouwer classical Comp complete computable structure Computer Science condition consistency construction continuum countable decision problem definition degree Department of Mathematics Dept discussion domain E-mail editors elementary equivalent Euler characteristic example existence Feferman field finite first-order formal formula foundational Fraenkel functions Godel Grothendieck ring hierarchy Hilbert infinite integer interpolation interpretation intuitionism intuitionistic isomorphic Journal of Symbolic language large cardinal axioms Lemma machine many-sorted Math mathematical logic Mathematical Society mathematicians measurable cardinals method modal logic model theory natural numbers nonstandard notion o-minimal ordinal P.O. Box paper Penelope Maddy Philos polynomial predicate principle probability probability amplitudes proof theory properties proposition proved quantifiers quantum computer question real numbers recursive relation semantics sentences sequence set theory subsets Symbolic Logic Tarski theorem type theory Univ University variables Woodin Woodin cardinals