Categoricity for an inferential $\omega$-logic and in $L_{\omega_1,\omega}$

John T. Baldwin; Constantin C. Brîncuş

Categoricity for an inferential $\omega$-logic and in $L_{\omega_1,\omega}$

Logic 2026-04-28 v2

Authors: John T. Baldwin , Constantin C. Brîncuş

Abstract

This paper provides two extensions of first order logic by ` $\omega$ -rules'. In each case we characterize the countable structures whose theory in the logic is categorical (has a unique model). In the one-sorted inferential $\omega$ -logic, both Robinson's system $Q$ and Peano Arithmetic become categorical. In the two-sorted generalized $\omega$ -logic we show each complete $L_{\omega_1,\omega}$ sentence defines the same class of structures as a first-order theory with the appropriate $G-\omega$ -rule. The results depend on proving that the inferential rules for the logics are categorical, i.e. they uniquely determine certain truth-conditions for the logical connectives and quantifiers.

Keywords

inferential $\omega$-logic $L_{\omega_1,\\omega}$ categoricity $\omega$-rules Robinson's system Q

Cite

@article{arxiv.2602.02854,
  title  = {Categoricity for an inferential $\omega$-logic and in $L_{\omega_1,\omega}$},
  author = {John T. Baldwin and Constantin C. Brîncuş},
  journal= {arXiv preprint arXiv:2602.02854},
  year   = {2026}
}

Comments

The original full version of this work has been divided into two separate papers. The present paper contains the technical results while a companion paper (Carnapian Frameworks and Categoricity of Arithmetic via Inferential $\omega$-logics) provides a philosophical discussion of these results

Categoricity for an inferential $\omega$-logic and in $L_{\omega_1,\omega}$

Abstract

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