The model \(N=\cup \{L[A]:\) A countable set of ordinals\(\}\) (Q1105592)

The class N is defined as the union of all classes L[A], where L is the constructible universe and A an arbitrary countable set of ordinals. It is shown that if N satisfies all the axioms of Zermelo-Fraenkel set theory (ZF), then N satisfies the Jensen Covering Property (for N instead of L). On the other hand, if N does not satisfy the covering property, then for every \(\alpha <\omega_ 1\), there is an inner model of ZF with \(\alpha\) measurable cardinals. It turns out that N is a model of ZF iff it is equal the Chang's model C. Some bounds on cardinalities of power sets P(\(\lambda)\) in N are given.