1995•Journal of Symbolic LogicRequires access

Scott incomplete Boolean ultrapowers of the real line

Masanao Ozawa

Abstract

Abstract An ordered field is said to be Scott complete iff it is complete with respect to its uniform structure. Zakon has asked whether nonstandard real lines are Scott complete. We prove in ZFC that for any complete Boolean algebra B which is not (ω, 2)-distributive there is an ultrafilter of B such that the Boolean ultrapower of the real line modulo is not Scott complete. We also show how forcing in set theory gives rise to examples of Boolean ultrapowers of the real line which are not Scott complete.

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Key concepts: Ultrafilter, Ultraproduct, Boolean algebras canonically defined, Stone's representation theorem for Boolean algebras, Real line, Complete Boolean algebra, Boolean algebra, Modulo