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On Rosser theories

Yong Cheng

TL;DR

This work generalizes Rosser theory beyond arithmetic languages to $n$-ary recursively enumerable relations by introducing $n$-Rosser, exact $n$-Rosser, effectively $n$-Rosser, and effectively exact $n$-Rosser theories via an interpretable numeral theory ${\sf Num}$. It develops a generalized Strong Double Recursion Theorem (SDRT) for vectorized relations and uses it to extend key Rosser-type results (including Putnam–Smullyan theorems) to all $n\ge 1$, proving equivalences among the four notions under local interpretations. The paper also extends the semi-$\sf DU$ and $\sf DU$ framework to $n$-ary relations, establishing a chain of implications that yields powerful meta-mathematical applications in arithmetic. Collectively, these results broaden the incompleteness meta-theory to a broad, arity-agnostic setting and clarify how definability of numeral and pairing structures interacts with Rosser properties.

Abstract

Rosser theories play an important role in the study of the incompleteness phenomenon and meta-mathematics of arithmetic. In this paper, we first define the notions of $n$-Rosser theories, exact $n$-Rosser theories, effectively $n$-Rosser theories and effectively exact $n$-Rosser theories (see Definition 1.6). Our definitions are not restricted to arithmetic languages. Then we systematically examine properties of $n$-Rosser theories and relationships among them. Especially, we generalize some important theorems about Rosser theories for recursively enumerable sets in the literature to $n$-Rosser theories in a general setting.

On Rosser theories

TL;DR

This work generalizes Rosser theory beyond arithmetic languages to -ary recursively enumerable relations by introducing -Rosser, exact -Rosser, effectively -Rosser, and effectively exact -Rosser theories via an interpretable numeral theory . It develops a generalized Strong Double Recursion Theorem (SDRT) for vectorized relations and uses it to extend key Rosser-type results (including Putnam–Smullyan theorems) to all , proving equivalences among the four notions under local interpretations. The paper also extends the semi- and framework to -ary relations, establishing a chain of implications that yields powerful meta-mathematical applications in arithmetic. Collectively, these results broaden the incompleteness meta-theory to a broad, arity-agnostic setting and clarify how definability of numeral and pairing structures interacts with Rosser properties.

Abstract

Rosser theories play an important role in the study of the incompleteness phenomenon and meta-mathematics of arithmetic. In this paper, we first define the notions of -Rosser theories, exact -Rosser theories, effectively -Rosser theories and effectively exact -Rosser theories (see Definition 1.6). Our definitions are not restricted to arithmetic languages. Then we systematically examine properties of -Rosser theories and relationships among them. Especially, we generalize some important theorems about Rosser theories for recursively enumerable sets in the literature to -Rosser theories in a general setting.
Paper Structure (13 sections, 42 theorems, 39 equations)

This paper contains 13 sections, 42 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Definitions of Rosser theories in the literature
  3. A new definition of Rosser theories
  4. Preliminary
  5. Some basic facts about Rosser theories
  6. A generalization of the Strong Double Recursion Theorem
  7. Generalizations of Theorem \ref{['effect Rosser']} to $n$-Rosser theories
  8. $n$-Rosser + strong definability of $n$-ary recursive functionals implies exact $n$-Rosser
  9. Semi-DU implies DU
  10. Some applications in meta-mathematics of arithmetic
  11. Equivalences under the definability of the paring function
  12. The second proof of Theorem \ref{['semi-DU is DU']}
  13. The third proof of Theorem \ref{['semi-DU is DU']}

Key Result

Theorem 1.2

Theorems & Definitions (115)

  • Definition 1.1: Smullyan93
  • Theorem 1.2: Smullyan93
  • Theorem 1.3: Putnam-Smullyan Theorem,Smullyan93
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 2.1: Translations and interpretations, Visser16, pp.10-13
  • Definition 2.2
  • Lemma 2.3: Separation Lemma, Smullyan93
  • ...and 105 more