Distinguishing three strong saturation properties in nonstandard analysis
The definitions of \({\kappa }\)-isomorphism property \((\text{IP}_{\kappa })\), \({\kappa }\)-special model axiom (\(\text{SMA}_{\kappa}\)), and full saturation were introduced by \textit{C. W. Henson} [``The isomorphism property in nonstandard analysis and its use in the theory of the Banach spaces'', J. Symb. Log. 39, 717-731 (1974; Zbl 0306.02055)] and \textit{D. Ross} [``Automorphisms of the Loeb algebra'', Fundam. Math. 128, 29-36 (1987; Zbl 0633.28010), ``The special model axiom in nonstandard analysis'', J. Symb. Log. 55, No. 3, 1233-1242 (1990; Zbl 0721.03049)] as generalizations of the \({\kappa}\)-saturation property. Full saturation is the strongest, then comes the \({\kappa}\)-special model axiom, and then the \({\kappa}\)-isomorphism property. Every time when a result is proved under a stronger property, the natural question is whether or not the result can be proved under a weaker property. So far there have been no examples in some mathematical field other than logic to show the different strengths among the strong saturation properties. The paper under review fills the gap. An example of a Loeb probability space is constructed which is a consequence of full saturation unprovable under \(\text{SMA}_{\kappa}\). It is shown that \(\text{SMA}_{{\aleph}_1}\) cannot be replaced by \(\text{IP}_{{\aleph}_1}\) in some theorem on Loeb spaces. Finally, it is proved that under \(\text{SMA}_{{\aleph}_1}\) every Loeb probability space has an ergodic (even strong mixing) transformation.
scientific article; zbMATH DE number 1340812
scientific article