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Some remarks to the axiom of prolongation. (English) Zbl 0624.03040

The axiom of prolongation plays an important role in the alternative set theory (AST). It describes a form of the saturation property. (Remember here the paper of C. W. Henson and H. J. Keisler [J. Symb. Logic 51, 377-386 (1986; Zbl 0624.03051)], where it is proved that the strength of nonstandardness in nonstandard theories is exactly dependent on the strength of the saturation axiom scheme.) It is proved in the paper that the assertion of the axiom of prolongation even in its weaker form cannot be extended (when saving the system of all classes) on cuts not being \(\sigma\)-classes. For it a transfinite construction, which forms to every cut I not being a \(\sigma\)-class a class function \(F: I\leftrightarrow^{1-1}I\) which is not a subclass of any set function, is used. Some quotations concerning the matter are given there, too.

MSC:

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

Citations:

Zbl 0624.03051
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