Cartagena Logic

TL;DR

Cartagena logic is a new infinitary logic with a clear syntax, extending ${ m L}_{"kappa

Contribution

It introduces generalized Boolean variables and a Boolean expansion of ${ m L}_{"kappa

Findings

Proves model-theoretic properties of Cartagena logic

Demonstrates expressive power through examples

Shows Cartagena logic approximates Shelah's ${ m L}^1_"kappa$

Abstract

We introduce a new kind of infinitary logic that we call Boolean expansion of ${\mathcal L}_{\kappa \kappa}$. This logic involves a new kind of variable, that we call generalised Boolean variable. These variables range over the powerset of a cardinal number in a way reminiscent of random variables. From this Boolean expansion, we extract a traditional infinitary logic, called Cartagena logic. We prove several model-theoretic properties of Cartagena logic, and give multiple examples of its expressive power. The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary ${\mathcal L}^1_\kappa$. The latter is not known to have a generative syntax, while Cartagena logic does have a very clear one.