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Making the Hyperreal Line Both Saturated and Complete

H. Jerome Keisler and James H. Schmerl
The Journal of Symbolic Logic
Vol. 56, No. 3 (Sep., 1991), pp. 1016-1025
DOI: 10.2307/2275069
Stable URL: http://www.jstor.org/stable/2275069
Page Count: 10
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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 $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, 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.

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