Mathematical Sciences: Topics in Logic and the Foundations of Mathematics

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Several research problems and problem areas are under investigation, primarily in proof theory, with applications to constructive and semi-constructive mathematics and theoretical computer science. In addition a new problem area is proposed, relating recursion theory, logic, and algebra. The topics are as follows: 1. Recursion-theoretic analogues between algebra and logic (e.g., Higman's finite embedding theorem and Kleene's finite axiomatizability theorem). 2. Proof-theoretic reduction of subsystems of analysis to weak theories. (a) Reduction to PRA and below. (b) Reduction to PA. 3. Proof-theory of constructive theories of functions and classes. (a) The strength of the general principle of monotone inductive definition. (b) The strength of a constructive programming system. (c) Polymorphism. The interplay between mathematical logic and theoretical computer science has always been and continues to be very fruitful.