The HOD Dichotomy

This paper provides a more accessible account of some of the material from Woodin [4] and [5]. All unattributed results are due to the first author. Recall that 0# is a certain set of natural numbers that codes an elementary embedding j : L → L such that j 6= id L. Jensen’s covering lemma says that if 0# does not exist and A is an uncountable set of ordinals, then there exists B ∈ L such that A ⊆ B and |A| = |B|. The conclusion implies that if γ is a singular cardinal, then it is a singular cardinal in L. It also implies that if γ ≥ ω2 and γ is a successor cardinal in L, then cf(γ) = |γ|. In particular, if β is a singular cardinal, then (β+)L = β+. Intuitively, this says that L is close to V . On the other hand, should 0# exist, if γ is an uncountable cardinal, then γ is an inaccessible cardinal in L. In this case, we could say that L is far from V . Thus, the covering lemma has the following corollary, which does not mention 0#.