Logic Seminar
John Baldwin
UIC
Categoricity for the inferential ω-logic and Lω1,ω
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.
Tuesday November 18, 2025 at 3:00 PM in 636 SEO