Covering Properties of Core Models

Summary

Theorem (Jensen's covering lemma) Assume 0^# does not exist. Let A be any uncountable set of ordinals. Then there is a B ∈ L such that B ⊇ A and card(B) = card(A).

In this paper, we outline Jensen's proof from a modern perspective. We isolate certain key elements of the proof which have become important both within and outside of inner model theory. This leads into an intuitive discussion of what core models are and the difficulties involved in generalizing Jensen's theorem to higher core models. Our hope is to give the reader some insight into these generalizations by concentrating on the simplest core model, L.

Jensen's theorem has striking consequences for cardinal arithmetic. Its conclusion implies that if ω₂ ≤ β and β is a successor cardinal of L, then cf(β) = card(β). In particular, if 0^# does not exist, then L computes successors of singular cardinals correctly. The covering lemma also implies that some of the combinatorial principles, which Jensen proved in L, really hold. (I.e., they hold in V.) For example, if 0^# does not exist and k is any singular cardinal, then □_K holds.

By an inner model, we mean a transitive proper class model of ZFC. If M is an inner model, then M has the covering property if for every uncountable set of ordinals A, there exists B ∈ M such that B ⊇ A and card(B) = card(A).