Journal of Symbolic Logic

Making the Hyperreal Line Both Saturated and Complete

H. Jerome Keisler and James H. Schmerl

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

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J. Symbolic Logic, Volume 56, Issue 3 (1991), 1016-1025.

First available in Project Euclid: 6 July 2007

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Keisler, H. Jerome; Schmerl, James H. Making the Hyperreal Line Both Saturated and Complete. J. Symbolic Logic 56 (1991), no. 3, 1016--1025. doi:10.2178/jsl/1183743748.

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