Universality properties of forcing
Francesco Parente, Matteo Viale
TL;DR
This paper investigates forcing as a method to realize universal models of initial-segment theories. It develops a framework based on Boolean-valued structures and good ultrafilters, enabling universality results via two forcing notions: the collapsing algebra and the stationary tower. The first result shows that, under an inaccessible $ obreak\delta$, any model of ${ m Th}_{ orall}(H_{ obreak\kappa^+}, ext{in}_{ obreak\Delta_0})$ of size $ obreak\le obreak\delta$ embeds into $H_{ obreak\check{ obreak ext{} obreak ext{} obreak ext{} obreak eta}^{ obreak ext{ } obreak} obreak}$; this yields universality for a Δ0-definable fragment. The second set of results extends universality to full theories with arbitrary relations using the stationary tower, together with large cardinal assumptions (notably Woodin cardinals) to ensure well-foundedness and accurate reflection of $H_ obreak ext{} obreak ext{} obreak$. Overall, the work highlights a universal aspect of forcing in model theory and provides new existence proofs for good ultrafilters on non-complete algebras, enabling robust embedding and representation theorems for forcing-constructed universes.
Abstract
The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.