Nonstandard models of arithmetic as an alternative basis for continuum considerations. (English) Zbl 0531.03045

Summary: From the point of view of 1) nonstandard models of arithmetic: A special type of strong cuts in the sense of Kirby and Paris are considered. It is proved that the pair of such a cut and the corresponding ground model may serve as a basis for an alternative construction of real numbers. Some other set theoretical properties are proved. 2) Nonstandard analysis: Using nonstandard methods a model for real numbers is constructed in a theory much weaker than Zermelo-Fraenkel set theory. 3) Alternative set theory: Considerations in a fragment of AST are made and a contribution to the shiftings of horizon problem is given.


03H15 Nonstandard models of arithmetic
03E70 Nonclassical and second-order set theories
03H05 Nonstandard models in mathematics
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