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Notes on Interpretability between Weak First-order Theories: Theories of Sequences

Lars Kristiansen, Juvenal Murwanashyaka

Abstract

We introduce a first-order theory $\mathsf{Seq}$ which is mutually interpretable with Robinson's $\mathsf{Q}$. The universe of a standard model for $\mathsf{Seq}$ consists of sequences. We prove that $\mathsf{Seq}$ directly interprets the adjuctive set theory $\mathsf{AST}$, and we prove that $\mathsf{Seq}$ interprets the tree theory $\mathsf{T}$ and the set theory $\mathsf{AST + EXT}$.

Notes on Interpretability between Weak First-order Theories: Theories of Sequences

Abstract

We introduce a first-order theory which is mutually interpretable with Robinson's . The universe of a standard model for consists of sequences. We prove that directly interprets the adjuctive set theory , and we prove that interprets the tree theory and the set theory .
Paper Structure (18 sections, 20 theorems, 52 equations, 6 figures)

This paper contains 18 sections, 20 theorems, 52 equations, 6 figures.

Table of Contents

  1. Introduction
  2. An overview of the paper.
  3. Discussion of some related work.
  4. Some considerations on text, alphabet strings and sequences.
  5. The theories $\mathsf{Seq}$ and $\mathsf{WSeq}$
  6. $\mathsf{Seq}$ is a sequential theory
  7. Indexed Sequences
  8. Terminology and Notation
  9. A Variant of $\mathsf{Seq}$
  10. $\mathsf{Seq}$ Plus Extras
  11. $\mathsf{Seq}$ interprets the tree theory $\mathsf{T}$
  12. The ideas
  13. Examples:
  14. Examples:
  15. Construction
  16. ...and 3 more sections

Key Result

theorem 1

$\mathsf{Q}$ interprets $\mathsf{Seq}$.

Figures (6)

  • Figure 1: The non-logical axioms of $\mathsf{Seq}$.
  • Figure 2: The non-logical axioms of $\mathsf{WSeq}$.
  • Figure 3: The non-logical axioms of $\mathsf{AST + EXT}$. The two first axioms are the axioms of $\mathsf{AST}$.
  • Figure 4: The non-logical axioms of $\mathsf{Seq}^*$.
  • Figure 5: The non-logical axioms of $\mathsf{Seq}^+$.
  • ...and 1 more figures

Theorems & Definitions (33)

  • theorem 1
  • lemma 1
  • proof
  • theorem 2: $\Sigma$-completeness of $\mathsf{WSeq}$
  • proof
  • lemma 2
  • proof
  • theorem 3
  • proof
  • corollary 1
  • ...and 23 more