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On the relationships between some meta-mathematical properties of arithmetical theories

Yong Cheng

TL;DR

The paper initiates a systematic map of how twelve metamathematical properties of arithmetical theories relate, focusing on whether one property implies another within RE theories and using $\mathbf{R}$ as a reference point. It proves that $\mathbf{R}$ possesses all twelve properties and that having a given property in a theory does not guarantee interpretability of $\mathbf{R}$. It then delineates a web of implications and non-implications among Rosser theories, EI, RI, TP, EHU, EU, Creative, $\mathbf{0}^{\prime}$, REW, RFD, RSS, and RSW, including preservation results under Boolean recursive isomorphisms and various constructive counterexamples. The results yield a detailed landscape of which metamathematical features strengthen or fail to strengthen others, clarifying limits on deducing interpretability or representation strength from weaker properties. The work provides a structured reference for future investigations into incompleteness and undecidability in arithmetic theories, with potential applications to meta-maxi-axiomatization and the design of theories with prescribed representational capacities.

Abstract

In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (Effectively inseparable), RI (Recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\mathbf{0}^{\prime}$ (theories with Turing degree $\mathbf{0}^{\prime}$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ of these properties, we examine whether the property $P$ implies $Q$.

On the relationships between some meta-mathematical properties of arithmetical theories

TL;DR

The paper initiates a systematic map of how twelve metamathematical properties of arithmetical theories relate, focusing on whether one property implies another within RE theories and using as a reference point. It proves that possesses all twelve properties and that having a given property in a theory does not guarantee interpretability of . It then delineates a web of implications and non-implications among Rosser theories, EI, RI, TP, EHU, EU, Creative, , REW, RFD, RSS, and RSW, including preservation results under Boolean recursive isomorphisms and various constructive counterexamples. The results yield a detailed landscape of which metamathematical features strengthen or fail to strengthen others, clarifying limits on deducing interpretability or representation strength from weaker properties. The work provides a structured reference for future investigations into incompleteness and undecidability in arithmetic theories, with potential applications to meta-maxi-axiomatization and the design of theories with prescribed representational capacities.

Abstract

In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (Effectively inseparable), RI (Recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, (theories with Turing degree ), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties and of these properties, we examine whether the property implies .
Paper Structure (7 sections, 49 theorems, 5 equations)

This paper contains 7 sections, 49 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. Relationships with interpreting the theory $\mathbf{R}$
  4. Relationships with Rosser theories
  5. Relationships with EI and RI theories
  6. Relationships with TP and EHU theories
  7. Relationships with other properties

Key Result

Lemma 2.9

Suppose $\sigma(x)=\exists y\, \phi_0(x,y)$ and $\sigma^{\prime}(x)=\exists y\, \phi_1(x,y)$ are two $\Sigma^0_1$-formulas with only one free variable. If $\mathfrak{N}\models \sigma^{\prime}(\overline{n})\leq \sigma(\overline{n})$ for some $n\in\mathbb{N}$, then $\mathbf{R}\vdash \neg(\sigma(\overl

Theorems & Definitions (111)

  • Definition 1.1: Rosser theories, the nuclei of a theory, EI theories
  • Definition 1.2: $\sf RI$, TP and EHU theories
  • Definition 1.3: EU, Creative, and $\mathbf{0}^{\prime}$ theories
  • Definition 1.4: REW, RFD, RSS and RSW theories
  • Remark 1.5
  • Definition 2.1: Basic Notation
  • Remark 2.2
  • Definition 2.3: Robinson Arithmetic $\mathbf{Q}$
  • Definition 2.5
  • Definition 2.6
  • ...and 101 more