Definability degrees for classes in the alternative set theory.

*(English)*Zbl 0646.03049The complexity of classes is investigated in the paper which is written in the framework of the alternative set theory. The complexity is related to the definability by positive formulas. The hierarchy arising by this kind of complexity has played a great role when inductive definitions have been investigated in classical mathematics. We think that the following results of the paper are the most interesting ones: i) Cuts (initial segments) on natural numbers are minimal in the mentioned hierarchy (for semisets also the opposite assertion holds, i.e., minimal semisets are equidefinable with suitable cuts). ii) A class X positively definable from a cut J can be obtained as \(X=R''J\) for a suitable set- definable relation R. iii) Every semiregular cut J is the least (in inclusion) of the system of equidefinable cuts. Another interesting circumstance is that the technical means of AST (conditioned by nonstandardness) may be used in places where the analogy to classical theory leads to complicated transfinite constructions.

Reviewer: K.Čuda

##### MSC:

03E70 | Nonclassical and second-order set theories |

03H15 | Nonstandard models of arithmetic |

03D70 | Inductive definability |