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Section: New Results

Computional interpretation of the axiom scheme of comprehension

The axiom scheme of comprehension is the cornerstone of second-order arithmetic, a logical theory in which most of mathematics can be formalized. Historically, comprehension was obtained from the negative translation of the axiom of choice, this axiom being interpreted by bar recursion. This led to cluttered and inefficient interpretations of second-order arithmetic.

Valentin Blot simplified this interpretation by proving that the axiom scheme of comprehension has a direct computational interpretation through a variant of bar recursion called update recursion. This new interpretation leads to a more efficient computational interpretation of proofs in second-order arithmetic, and paves the way for a convergence of the two existing interpretations: bar recursion and System F.