Endomorphic cuts and tails.

*(English)*Zbl 0634.03053The universe of AST (Alternative Set Theory) is not rigid as it is in classical set theory. There are nonidentical automorphisms (and even endomorphisms) of the universal class in AST. An endomorphic image of the universal class (called an endomorphic universe) is “the whole mathematical world” in small, which is embedded to the “original world”. Considerations of this tuple of “worlds” are similar to considerations of a structure and its enlargement in nonstandard analysis. Due to endomorphic universes two remarkable types of cuts on the class N (of natural numbers, also infinitely large) appear. (Sometimes a cut may be of both of the types.) One of them consists of the traces of transitive endomorphic universes of N, we call them endomorphic cuts. The second ones are an analogy to enlargements of the set of natural numbers; they are defined by \(E_ A(FN)\) and are called here endomorphic tails. We study the separability of infinitely large natural numbers by these cuts, which is a characterization of a large distance. This fact is confirmed, e.g., by the following equivalent of the separability: Natural numbers \(\alpha <\beta\) are separable by an endomorphic tail ((\(\exists J\) end. tail) (\(\alpha\in J\subset \beta))\) iff there is an infinitely large natural number \(\gamma\) such that \(\alpha <\gamma \leq \beta\) and every infinitely large natural number definable from \(\gamma\) is larger than \(\alpha\).

Reviewer: K.Čuda