The model N = ∪ { L[A]: A countable set of ordinals}

Abstract

This paper continues the study of covering properties of models closed under countable sequences. In a previous article we focused on C. Chang's Model (the equivalent of the Constructible Universe for the language L ω1 ω1 ). Our purpose is now to deal with the model N = ∪ { L[A]: A countable ⊂ Ord}. We study here relations between covering properties, satisfaction of ZF by N, and cardinality of power sets. Under large cardinal assumptions (existence of ω 1 measurable cardinals or of a compact cardinal) N is strictly included in Chang's Model C, it may thus be interesting to obtain an analogue for N of the covering result for C. As in [13], we say that N satisfies the covering property if, for any set of ordinals X in the Universe, there exists a set Y in N such that X ⊂ Y and |Y| = |X| ℵ0 . In the first part, we show that if N does not satisfy the covering property, then for any α < ω 1, there exists an inner model with α measurable cardinals. Throughout this paper, we compare three situations: 1. (b) V is the universe constructible from a countable sequence of ultrafilters on different measurable cardinals. 2. (b) N does not satisfy the covering property. 3. (c) There exist ω 1 measurable cardinals or a compact cardinal. Section 2 is devoted to the relation between satisfaction of ZF by N and the covering property. If (a) holds, then N satisfies both ZF and the covering property. On the contrary, in case (c), N satisfies neither ZF nor the covering property. It is thus natural to obtain that the satisfaction of ZF by N implies the satisfaction of the covering property. Finally we study the cardinality of P( λ) ∩ N for a cardinal λ in the different cases. For example we show that if N does not satisfy the covering property, then (2 ℵ0) + is inaccessible in any L[ A] where A is a countable set of ordinals. This is not the case if (a) holds. Let the expression “ λ + is inaccessible in N” mean that for any α < λ + | P( α) ∩ N| ⩽ λ. If situation (c) occurs, then (in analogy with 0 †) there exists a measurable cardinal κ such that “ κ + is inaccessible in N”. The large cardinal assumption is necessary since we show that the last fact implies the violation of the covering property by N.