Sochor, A. Complexity of the axioms of the alternative set theory. (English) Zbl 0792.03037 Commentat. Math. Univ. Carol. 34, No. 1, 33-45 (1993). The main result of the paper (and in my opinion a very nice one) asserts that if \(T\) is a complete theory with set language stronger than \(\text{ZF}_{\text{fin}}\) such that the axiom of extensionality for classes \(+\;T\;+\) \((\exists X)\Phi_ i(X)\) is consistent for \(1\leq i\leq k\) (each alone), where \(\Phi_ i\) are normal formulas, then \(\text{AST}+ T+ (\exists X)\Phi_ 1+\cdots+ (\exists X)\Phi_ k\;+\) scheme of choice is consistent (AST being alternative set theory). Using this result, the author proves another nice result that there is no proper \(\Delta_ 1\)- formula in AST + scheme of choice. If for some normal formulas \(\Phi_ 1(X)\Phi_ 2(X)\) we have AST + scheme of choice \(\lvdash(\exists X)\Phi_ 1(X)\equiv (\forall X)\Phi_ 2(X)\), then there is a set formula \(\Psi\) such that the equivalence of \(\Psi\) and both mentioned formulas can be proved in this theory. The result is expressed in a stronger version.In the first part of the paper, the author discusses the complexity of the axioms of AST. He proves, e.g., that all the three substantial axioms of AST (prolongation, choice and cardinalities) can be expressed as both \(\Sigma_ 2\) and \(\Pi_ 2\) formulas in AST without the mentioned axiom (even stronger versions are given). Using the first mentioned result, the author proves that the three mention axioms cannot be expressed as \(\Pi_ 1\) formulas. Axioms of prolongation and cardinalities cannot be expressed as \(\Sigma_ 1\) formulas (which is proved by other means).The paper is written comprehensibly, but, except for the common logical means, it uses also advanced techniques of AST, e.g., revealment. Reviewer: K.Čuda (Praha) MSC: 03E70 Nonclassical and second-order set theories 03H15 Nonstandard models of arithmetic 03A05 Philosophical and critical aspects of logic and foundations 03E35 Consistency and independence results 03D55 Hierarchies of computability and definability Keywords:complexity of formulas; \(\Pi_ 2\)-formula; extension of axiomatic system; prolongation; alternative set theory; choice; cardinalities × Cite Format Result Cite Review PDF Full Text: EuDML