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  • Print publication year: 2012
  • Online publication date: December 2012

13 - The HOD Dichotomy


The sixteenth Appalachian Set Theory workshop was held at Cornell University in Ithaca on April 7, 2012. The lecturer was Hugh Woodin. As graduate students Jacob Davis and Daniel Rodríguez assisted in writing this chapter, which is based on the workshop lectures.


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 : LL such that j ≠ id ↾ L. Jensen's covering lemma says that if 0# does not exist and A is an uncountable set of ordinals, then there exists BL such that AB 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#.