Tag Archives: HOD Dichotomy

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

In light of your HOD Dichotomy I interpreted the HOD conjecture to say that if there is an extendible delta then HOD correctly computes successors of singulars above \delta correctly. All I meant was that if you drop the extendible then this conclusion need not hold. I am guessing (I really don’t know) that if there is an extendible then this conclusion does hold (and hence the HOD Conjecture is true).

Unless you can derive extendibles from some form of maximality the consequence I would draw from the HOD conjecture would be that maximality violates the existence of extendible cardinals.

Best, Sy

Re: Paper and slides on indefiniteness of CH

Dear Bob,

I guess I have used it both ways. But also I am most interested in (A) but in the form ZFC + extendible proves the formal statement of the HOD Conjecture i.e. that there is a proper class of regular cardinals which are not \omega-strongly measurable in HOD.

I suppose I should have called the formal statement of the HOD Conjecture, the HOD Hypothesis; and then defined the HOD Conjecture as the conjecture that ZFC (or ZFC + extendible) proves the HOD Hypothesis. Probably it is too late to make that change.

Here is a simple version the HOD Dichotomy theorem:

Theorem. Suppose \delta is extendible. Then the following are equivalent.

  1. HOD Hypothesis.
  2. There is a regular cardinal above \delta which is not \omega-strongly measurable in HOD.
  3. There is a regular cardinal above \delta which is not measurable in HOD.
  4. For every singular cardinal \gamma > \delta, \gamma is singular in HOD and \gamma^+ is the \gamma^+ of HOD.
  5. \delta is supercompact in HOD witnessed by the restriction to HOD of supercompactness measures in V.

Regards, Hugh

Re: Paper and slides on indefiniteness of CH

Dear all,

Here is some background for those who are interested. My apologies to those who are not, but delete is one key stroke away.

Jensen’s Covering Theorem states that V is either very close to L or very far from L. This opens the door for consideration of 0^\# and current generation of large cardinal axioms.

Details: “close to L” means that L computes the successors of all singular (in V) cardinals correctly. “far from L” means every uncountable cardinal is inaccessible in L.

The HOD Dichotomy Theorem [proved here] is in some sense arguably an abstract generalization of Jensen’s Covering Theorem. This theorem states that if there is an extendible cardinal then V is either very close to \text{HOD} or very far from \text{HOD}.

Details: Suppose \delta is an extendible cardinal. “very close to \text{HOD}” means the successor of every singular cardinal above \delta is correctly computed by \text{HOD}. “very far from \text{HOD}” means that every regular cardinal above \delta is a measurable cardinal in \text{HOD} and so \text{HOD} computes no successor cardinals correctly above \delta.

Aside: The restriction to cardinals above \delta is necessary by forcing considerations and the close versus far dichotomy is much more extreme than just what is indicated above about successor cardinals.

The pressing question then is: Is the \text{HOD} Dichotomy Theorem really a “dichotomy” theorem?

The \text{HOD} Conjecture is the conjecture that it is not; i.e. if there is an extendible cardinal then \text{HOD} is necessarily close to V.

Given set theoretic history, arguably the more plausible conjecture is that \text{HOD} Dichotomy Theorem is a genuine dichotomy theorem and so just as 0^\# initiates a new generation of large cardinal axioms (that imply V is not L) there is yet another generation of large cardinal axioms which corresponds to the failure of the \text{HOD} Conjecture.

But now there is tension with the Inner Model Program which predicts that \text{HOD} Conjecture is true (for completely unexpected reasons).

My question to Sy was implicitly: Why does he not, based on maximality, reject \text{HOD} Conjecture since disregarding the evidence from the Inner Model Program, the most natural speculation is that the \text{HOD} Conjecture is false.

The point here is that the analogous conjecture for L is false (since 0^\# exists).

So one could reasonably take the view that the \text{HOD} Conjecture is as misguided now as would have been the conjecture that L is close to V given the Jensen Covering Theorem. (Let’s revise history and pretend that Jensen’s Covering Theorem was proved before measurable cardinals etc. had been defined and analyzed).