Abstract

Summary. The goal is show that the reflection theorem holds. The theorem is as usual in the Morse-Kelley theory of classes (MK). That theory works with universal class which consists of all sets and every class is a subclass of it. In this paper (and in another Mizar articles) we work in Tarski-Grothendieck (TG) theory (see [15]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [13] and some concepts of MK are modeled. The concepts are: the class On of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of non-empty elements of universe, etc. The reflection theorem states that if A ξ is an increasing and continuous transfinite sequence of non-empty sets and class A = � ξ∈On A ξ, then for every formula H there is a strictly increasing continuous mapping F: On → On such that if κ is a critical number of F (i.e. F(κ) = κ> 0) and f ∈ A VAR κ, then A, f | = H ≡ Aκ, f | = H. The proof is based on [11]. Besides, in the article it is shown that every universal class is a model of ZF set theory if ω (the first infinite ordinal number) belongs to it. Some propositions concerning ordinal numbers and sequences of them are also present.

Citations

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