The Philosophy of Mathematics

Alfred Tarski arbitrary argument arithmetic axiom of choice axiom system basic classical completeness theorem concepts consider consistent constants construction contains corresponding definable definition denote discussion domain elementary equivalent existence expression fact finite first-order formal derivability formal system formula free occurrences geometry Georg Kreisel given Gödel number hence hyperarithmetic hypothesis impredicative individual infinite infinitesimal input intuitionistic Intuitionistic Logic Journal of Symbolic Kreisel Kurt Gödel left column Leibniz mathe means metamathematical method natural numbers non-standard Non-standard Analysis obtained ordered field ordinal paper particular Peano axioms Philosophy of Mathematics premisses problem procedure proof properties proved quantifiers ramified analysis real closed field real numbers recursive function recursively enumerable relation rules of inference S₁ satisfiable second-order Section semantic entailment semantic tableau sequence set of cwffs set theory suitable counter-example Symbolic Logic transfinite true U₁ valid