Hugh Woodin at Cornell University on April 7, 2012

Appalachian set theory

Saturday, April 7, 2012

Cornell University

Hugh Woodin : "The HOD dichotomy"

Participants in this workshop

Lecture notes from this workshop by Hugh Woodin, Jacob Davis and Daniel Rodriguez (PDF)

Workshop description

A fundamental theorem in Set Theory is Jensen's Covering Lemma. This lemma is in essence a dichotomy theorem: either V is very close to L or V is very far from L. For example, one of the following must hold:

  1. For every singular cardinal λ, λ is a singular cardinal in L and L correctly computes λ+.
  2. For every singular cardinal λ, λ is not a singular cardinal in L and L does not correctly compute λ+.

Assuming that there is an extendible cardinal then there is a version of this dichotomy theorem which holds with L replaced by HOD. This is the HOD Dichotomy Theorem which is principal subject for this workshop. The techniques involve analyzing stationary sets and their relationships to HOD. This all leads to the HOD Conjecture which informally is the conjecture that HOD Dichotomy is not a dichotomy at all and that assuming there is an extendible cardinal, HOD must be close to V. This is related to the possible existence of an ultimate version of L.

The workshop will present the proof of the HOD Dichotomy Theorem with minimal prerequisites. Some connections with the inner model problem for one supercompact cardinal will also be discussed as well as implications for the Axiom of Choice.


Participant travel support

Funds provided by the NSF will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and postdocs, and to faculty who do not old federal research grants. Please request such funds now by sending the following information to Ernest Schimmerling by email.