On the Exhaustion by the Segment and Some Generalizations of This Concept
The study is carried out in axiomatic of the alternative set theory, and some generalizations of σ- and π-classes concepts are defined. We fix some new horizon A which is larger than the initial segment of all finite natural numbers. Further we consider all unions and intersections of set-theoretic classes families being indexed by the elements of A. It is shown, that these generalizations of the concepts preserve basic properties of σ- and π-classes. In particular, if some class is an intersection and a union of the mentioned families simultaneously then it is a set-theoretic class. The problem of finding all of N segments exhausted by every correspondent subsequent subsegment which was set up in one of the earlier works of the author is solved as an application of the obtained results. It turns out that the only segment with such properties is the segment of all finite natural numbers. As a conclusion, we demonstrate that the construction of all measurable classes family full analogue closed with a respect to arbitrary intersection and unions and using a larger horizon as a main one, is impossible.