If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...

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 . 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 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. V is the universe constructible from a countable sequence of ultrafilters on different measurable cardinals. 2. N does not satisfy the covering property. 3. 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 holds, then N satisfies both ZF and the covering property. On the contrary, in case , 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 ∩ N for a cardinal λ in the different cases. For example we show that if N does not satisfy the covering property, then + is inaccessible in any L [ A ] where A is a countable set of ordinals. This is not the case if holds. Let the expression “ λ + is inaccessible in N ” mean that for any α λ + | P ∩ N | ⩽ λ . If situation occurs, then 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. (shrink)