Consequences of the Reflection Theorem
Grzegorz Bancerek · 1990
Some consequences of the reflection theorem are discussed. To formulate them the notions of elementary equivalence and subsystems, and of models for a set of formulae are introduced. Besides, the concept of cofinality of a ordinal number with second one is used. The consequences of the reflection theorem (it is sometimes called the Scott-Scarpellini lemma) are: (i) If Aξ is a transfinite sequence as in the reflection theorem (see [9]) and A = � ξ∈On Aξ, then there is an in-creasing and continuous mapping φ from On into On such that for every critical number κ the set Aκ is an elementary subsystem of A (Aκ ≺ A). (ii) There is an increasing continuous mapping φ: On → On such that Rκ ≺ V for each of its critical numbers κ (V is the universal class and On is the class of all ordinals belonging to V). (iii) There are ordinal numbers α cofinal with ω for which Rα are models of ZF set theory. (iv) For each set X from universe V there is a model of ZF M which belongs to V and has X as an element.