Quantified universes and ultraproducts

✍ Scribed by Alireza Mofidi; Seyed-Mohammad Bagheri

• Book ID: 102488192
• Publisher: John Wiley and Sons
• Year: 2012
• Tongue: English
• Weight: 177 KB
• Volume: 58
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.201110019

✦ Synopsis

Abstract

A quantified universe is a set M equipped with a Riesz space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}_n$\end{document} of real functions on M^n^, for each n, and a second order operation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I:\mathcal {A}\rightarrow \mathbb R$\end{document}. Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed under this construction.