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Some automorphisms of natural numbers in the alternative set theory. (English) Zbl 0583.03042
In the paper a method is described for proving the existence of nontrivial automorphisms of the class of all natural numbers N (and hence also of the universal class V) with special properties. We may ask if such an automorphism is identical on an initial segment (up to an infinite natural number). We can also ask if the automorphism (a) majorizes, or (b) minorizes an arbitraryly chosen countable system of definable functions, or (c) majorizes or minorizes, alternatively, two countable systems of definable functions on sufficiently large intervals of infinite natural numbers. The existence of automorphisms is proved using suitable theories which have models being elementary substructures of N. Then, roughly speaking, ultrapowers of these models are isomorphic with N and automorphisms of ultrapowers generate automorphisms of N. For exact formulations and constructions, techniques of AST are used. The existence of initial models depends on some properties of strong indiscernibles. The paper substantially extends the former fact that there are nontrivial automorphisms being identical up to some infinite natural number.
Reviewer: K.Čuda
MSC:
03E70 Nonclassical and second-order set theories
03C50 Models with special properties (saturated, rigid, etc.)
03H15 Nonstandard models of arithmetic
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