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Combinatorial principles in nonstandard analysis. (English) Zbl 1016.03070

One of the more significant concepts within nonstandard analysis is the notion of \(\kappa\)-saturation. Associated with \(\kappa\)-saturation is the Henson isomorphism property \(\text{IP}(\kappa)\). Another saturation principle is the special model axiom \(\text{SMA}(\kappa).\) The first author of this paper initiated the study of combinatorial principles \(\Delta_0\) and \(\Delta_1\) where these might replace \(\text{IP}(\kappa)\) or \(\text{SMA}(\kappa)\) for \(\kappa = \aleph_0.\) Indeed, \(\text{IP}(\aleph_0)\) if and only if \(\Delta_1.\) In this paper, the authors continue this sort of investigation using superstructure Robinson-styled nonstandard analysis. They give equivalent formulations for \(\Delta_0\), \(\Delta_1\) and generalize them so that they imply \(\kappa\)-saturation for any cardinal \(\kappa.\) They show that many results previously established from \(\text{IP}(\kappa)\) follow from weaker assumptions. They then show how to use these combinatorial principles in practical work.

MSC:

03H05 Nonstandard models in mathematics
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