A Limiting Rst Order Realizability Interpretation

Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit. Computation in the limit, or more formally, limiting recursion, is a central notion of learning theory by Gold and Putnam 9, 21, 18]. We will show that a realizability interpretation via limiting recursive functions is a natural modeling of LCM for rst order arithmetic. We will point out that this will enable automatic extraction of limit-algorithms from some classical proofs of well-known transsnite theorems, e.g., Hilbert's original proof of his famous nite basis theorem, once blamed as \theology" by P. Gordan.