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Robinson Splitting Theorem and $Σ_1$ Induction

Yong Liu, Cheng Peng, Mengzhou Sun

Abstract

The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}Σ_1$, with lowness replaced by superlowness.

Robinson Splitting Theorem and $Σ_1$ Induction

Abstract

The Robinson Splitting Theorem states that a c.e. degree splits over any low c.e. degree . We prove that a weaker version of this theorem holds in models of , with lowness replaced by superlowness.
Paper Structure (12 sections, 20 theorems, 18 equations)

This paper contains 12 sections, 20 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:sacks']}
  4. A single requirement
  5. A block of requirements
  6. Dynamic priority assignments
  7. The construction and verification
  8. Proof of Theorem \ref{['thm:main']}
  9. Single requirement
  10. Block of requirements
  11. Construction and verification
  12. Discussion

Key Result

Theorem 1.1

The following holds in models of $\mathsf{P}^-+\mathsf{I}\Sigma_1$: Let $\bm{b}$ be a c.e. degree. Then there exist incomparable c.e. degrees $\bm{a_0}$ and $\bm{a_1}$ such that $\bm{b}= \bm{a_0}\vee \bm{a_1}$.

Theorems & Definitions (31)

  • Theorem 1.1: Sacks Splitting Theorem RN158RN82
  • Theorem 1.2: Robinson's Splitting Theorem RN129
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Lemma 2.2: H. Friedman
  • Definition 2.3
  • Lemma 2.4: RN384
  • Lemma 2.5
  • proof
  • ...and 21 more