Making the Hyperreal Line Both Saturated and Complete
H. Jerome Keisler, James H. Schmerl · Journal of Symbolic Logic · 1991
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.