A saturation property of structures obtained by forcing with a compact family of random variables

TL;DR

This paper demonstrates that structures built via forcing with a compact family of random variables exhibit a saturation property for existential types, with limitations for universal types, advancing the understanding of Boolean-valued models in arithmetic.

Contribution

It establishes a saturation property for structures obtained from compact families of random variables, clarifying their model-theoretic behavior in bounded arithmetic.

Findings

Structures are saturated for existential types under compactness.

Saturation does not extend to universal types.

Provides a concrete example illustrating the limitation.

Abstract

A method how to construct Boolean-valued models of some fragments of arithmetic was developed in Krajicek (2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random variables defined on a pseudo-finite sample space. We show that under a fairly natural condition on the family (called compactness in K.(2011)) the resulting structure has a property that is naturally interpreted as saturation for existential types. We also give an example showing that this cannot be extended to universal types.