John Baldwin : Categoricity for the inferential ω-logic and Lω1,ω

Posted by Matthew Harrison-Trainor , part of the Logic Seminar.

At

Nov. 18, 2025, 3 p.m.

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Abstract

This paper provides two extensions of first order logic by ‘ω-rules’. In each case we characterize the countable structures that have a categorical theory. In the one- sorted inferential ω-logic, both Robinson’s system Q and Peano Arithmetic become cat- egorical. In the two-sorted generalized ω-logic we show each Lω1,ω sentence defines the same class of structures as a first-order theory with the appropriate ω-rule. These logics are much weaker than any other proposed argument for the categoricity of arithmetic. 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.