On Measure Quantifiers in First-Order Arithmetic (Long Version)
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2021
- DOI:
- 10.48550/arXiv.2104.12124
- arXiv:
- arXiv:2104.12124
- Bibcode:
- 2021arXiv210412124A
- Keywords:
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- Computer Science - Logic in Computer Science;
- Mathematics - Logic;
- F.1.1;
- F.1.2;
- F.4.1