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Israel Journal of Mathematics

, Volume 207, Issue 1, pp 449–478 | Cite as

On the structure of the degrees of relative provability

  • Uri Andrews
  • Mingzhong Cai
  • David Diamondstone
  • Steffen Lempp
  • Joseph S. Miller
Article

Abstract

We investigate the structure of the degrees of provability, which measure the proof-theoretic strength of statements asserting the totality of given computable functions. The degrees of provability can also be seen as an extension of the investigation of relative consistency statements for first-order arithmetic (which can be viewed as Π 1 0 -statements, whereas statements of totality of computable functions are Π 2 0 -statements); and the structure of the degrees of provability can be viewed as the Lindenbaum algebra of true Π 2 0 -statements in first-order arithmetic. Our work continues and greatly expands the second author’s paper on this topic by answering a number of open questions from that paper, comparing three different notions of a jump operator and studying jump inversion as well as the corresponding high/low hierarchies, investigating the structure of true Π 1 0 -statements as a substructure, and connecting the degrees of provability to escape and domination properties of computable functions.

Keywords

Turing Machine Computable Function Transition Stage Total Function Minimal Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    M. Cai, Degrees of relative provability, Notre Dame Journal of Formal Logic 53 (2012), 479–489.zbMATHMathSciNetCrossRefGoogle Scholar
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Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  • Uri Andrews
    • 1
  • Mingzhong Cai
    • 2
  • David Diamondstone
    • 3
  • Steffen Lempp
    • 1
  • Joseph S. Miller
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Dartmouth CollegeHanoverUSA
  3. 3.GoogleMountain ViewUSA

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