Models of AST without choice. (English) Zbl 0561.03028

Čuda, K.; Vojtášková, B.

Commentat. Math. Univ. Carol. 25, 555-589 (1984).

Two interpretations in the alternative set theory (AST) are constructed in the paper. They prove the independence of the axiom of choice and even of the axiom of weak choice of the other axioms of AST. Moreover, the method of the construction of the interpretations may be quite remarkable for those which are interested in nonstandard models of arithmetic. The method improves nontrivially (especially in the second interpretation) the method from A. Vencovská’s paper: Independence of the axiom of choice in the alternative set theory [Open days in model theory and set theory, Proceedings of a Conference held in September 1981 at Jadwisin; W. Guzicki, W. Marek, A. Pelc, C. Rauszer (eds.) (Leeds, 1984)], where for the proof of the independence of the axiom of choice an interpretation in a stronger theory is used.

MSC:

03E70	Nonclassical and second-order set theories
03E35	Consistency and independence results
03E25	Axiom of choice and related propositions

Keywords:

basic equivalence; fully revealed class; endomorphic universe; standard extension; ultraproduct; alternative set theory; axiom of choice; weak choice