Pathwise-random trees and models of second-order arithmetic

George Barmpalias; Wei Wang

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arXiv · 2104.12066

Pathwise-random trees and models of second-order arithmetic

George Barmpalias·Wei Wang

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Abstract

A tree is pathwise-random if all of its paths are Martin-Lof random. We show that (a) no weakly 2-random real computes a perfect pathwise-random tree; it follows that the class of perfect pathwise-random trees is null, with respect to any computable measure; (b) there exists a positive-measure pathwise-random tree which does not compute any complete extension of Peano arithmetic; and (c) there exists a perfect pathwise-random tree which does not compute any tree of positive measure and finite randomness deficiency. We then obtain models of second-order arithmetic that separate compactness principles below weak Konigs lemma, answering questions by Chong et al.(2019).

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BibTeX RIS

George Barmpalias, Wei Wang. 2021-04-25. Pathwise-random trees and models of second-order arithmetic. https://arxiv.org/abs/2104.12066

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