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Making the hyperreal line both saturated and complete

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler  and
James H. Schmerl
H. Jerome Keisler
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
James H. Schmerl
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
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Abstract

In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 1016 - 1025
Copyright
Copyright © Association for Symbolic Logic 1991

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