Pathwise-random trees and models of second-order arithmetic
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909207942397952 |
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| author | Barmpalias, George Wang, Wei |
| author_facet | Barmpalias, George Wang, Wei |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_12066 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Pathwise-random trees and models of second-order arithmetic Barmpalias, George Wang, Wei Logic 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). |
| title | Pathwise-random trees and models of second-order arithmetic |
| topic | Logic |
| url | https://arxiv.org/abs/2104.12066 |