×

Constructibility and shiftings of view. (English) Zbl 0583.03040

The most striking result of this paper is such an interpretation of [AST \(+\) strong scheme of choice] in AST which preserves sets and \(\in\). (Moreover a theory in which there is a formula \(\phi\) (X,Y) determining a well-ordering of classes is interpreted.) It is obtained by adapting the constructive process for AST. The matter is investigated in a quite exhaustive manner and there are a lot of other interesting results here. Let us mention one of them. If a well-ordering of V of type \(\Omega\) is coded in a class Q then the following are equivalent: a) There is a well- ordering \(<\) such that the classes constructed from Q on stages less than \(<\) form the system of Q-constructible classes (they are closed under the constructible process). b) There is a codable system of classes \({\mathfrak M}\) with Q, FN\(\in {\mathfrak M}\) determining an interpretation of the formalization of AST (i.e. an interpretation of all formal formulas being finite sets and axioms of AST). The interpretations preserving sets, \(\in\) and FN (thus restricting only the system of classes), called restrictions of view, are also considered. Let us still mention that the author has recently constructed an interpretation of a very strong form of AST in the weakest possible theory using another adaptation of the constructible process (sets and \(\in\) are not preserved).
Reviewer: K.Čuda

MSC:

03E70 Nonclassical and second-order set theories
03E45 Inner models, including constructibility, ordinal definability, and core models
03H15 Nonstandard models of arithmetic
03E25 Axiom of choice and related propositions
03E35 Consistency and independence results