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arXiv · 2104.12124

On Measure Quantifiers in First-Order Arithmetic (Long Version)

Melissa Antonelli·Ugo Dal Lago·Paolo Pistone
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Abstract

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space.

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Melissa Antonelli, Ugo Dal Lago, Paolo Pistone. 2021-04-25. On Measure Quantifiers in First-Order Arithmetic (Long Version). https://arxiv.org/abs/2104.12124

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