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Pathwise-random trees and models of second-order arithmetic

George Barmpalias, Wei Wang

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).

Pathwise-random trees and models of second-order arithmetic

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).

Paper Structure

This paper contains 17 sections, 24 theorems, 69 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our results on pathwise randomness
  3. Our results on calibrating compactness principles
  4. Background and recent work on models of compactness
  5. Algorithmic randomness and trees
  6. Random reals and pathwise-random trees: Theorem \ref{['G6z9r7C93e']}
  7. Hypergraphs from computations of arrays
  8. Array-functionals and pathwise-randomness
  9. Avoiding positive trees: proof of Theorem \ref{['uZbmsMF7A6']}
  10. Forcing conditions and Turing functionals
  11. Hitting-sets and regularity
  12. Forcing convergence or divergence
  13. The model for (b) of Theorem \ref{['rms6rS57S4']}
  14. Positive trees not computing PA: Theorem \ref{['xJFhzQCXsu']}
  15. Patey's theorem
  16. ...and 2 more sections

Key Result

Theorem 1.2

If $z$ is weakly 2-random:

Theorems & Definitions (49)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9: Models
  • ...and 39 more