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Quantified universes and ultraproducts
Author(s) -
Mofidi Alireza,
Bagheri SeyedMohammad
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201110019
Subject(s) - ultraproduct , universe , mathematics , metric (unit) , space (punctuation) , order (exchange) , basis (linear algebra) , set (abstract data type) , metric space , pure mathematics , combinatorics , physics , computer science , astrophysics , geometry , philosophy , linguistics , operations management , finance , economics , programming language
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.