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Descending sequences in reflection hierarchies

Mateusz Łełyk, James Walsh

TL;DR

This work analyzes when descending sequences of n-consistent r.e. theories, each proving the n-consistency of the next, exist under different encodings and definability regimes. It distinguishes slice and index encodings, and uniform versus non-uniform interpretations, to map out positive constructions and negative barriers across all m, highlighting that encoding choice and definability level critically shape feasibility. Techniques such as Visser's descending-tower construction and Craig's trick are employed to produce sequences and prove nonexistence results, including extensions to 0'-recursive contexts. The results clarify how reflection principles and arithmetical encodings interact, revealing a nuanced landscape of possible versus impossible descent sequences in reflection hierarchies. Overall, the paper delineates precise boundaries for the existence of Γ-definable, n-consistent sequences under various encodings and base theories, advancing our understanding of uniformity, definability, and meta-mlogical strength in arithmetic.

Abstract

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to $0'$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $Γ$ are there $Γ$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $Γ$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.

Descending sequences in reflection hierarchies

TL;DR

This work analyzes when descending sequences of n-consistent r.e. theories, each proving the n-consistency of the next, exist under different encodings and definability regimes. It distinguishes slice and index encodings, and uniform versus non-uniform interpretations, to map out positive constructions and negative barriers across all m, highlighting that encoding choice and definability level critically shape feasibility. Techniques such as Visser's descending-tower construction and Craig's trick are employed to produce sequences and prove nonexistence results, including extensions to 0'-recursive contexts. The results clarify how reflection principles and arithmetical encodings interact, revealing a nuanced landscape of possible versus impossible descent sequences in reflection hierarchies. Overall, the paper delineates precise boundaries for the existence of Γ-definable, n-consistent sequences under various encodings and base theories, advancing our understanding of uniformity, definability, and meta-mlogical strength in arithmetic.

Abstract

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the -consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to -recursive sequences. We consider a general version of this problem: For arbitrary , for which complexity classes are there -definable sequences of -consistent r.e. theories each of which proves the -consistency of the next? The answer to this question depends not only on and but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.

Paper Structure

This paper contains 20 sections, 11 theorems, 55 equations, 1 table.

Table of Contents

  1. Introduction
  2. Encoding Sequences
  3. A Summary of the Main Results
  4. Outline of the Paper
  5. Preliminaries
  6. Reflection Principles
  7. The Index Encoding
  8. A Uniform Interpretation
  9. A Non-uniform Interpretation
  10. The Slice Encoding
  11. The Slice Encoding
  12. Visser's Technique
  13. The General Case
  14. Craig's Trick
  15. Generalizing a Negative Result
  16. ...and 5 more sections

Key Result

theorem 1.1

There is a recursive sequence $(T_n)_{n\in\mathbb{N}}$ of sound r.e. extensions of $\mathsf{PA}$ such that for each $n\in\mathbb{N}$, $T_n \vdash \mathsf{Con}(T_{n+1}).$ However, there is no recursively enumerable sequence $(T_n)_{n\in\mathbb{N}}$ of consistent r.e. extensions of $\mathsf{PA}$ such

Theorems & Definitions (47)

  • theorem 1.1: H. Friedman, Smoryński, Solovay
  • theorem 1.2: Pakhomov--Walsh
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 37 more