HOD dichotomy

The HOD dichotomy theorem is a theorem which shows that HOD, the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds:

• Every uncountable cardinal is inaccessible in \(L\).
• For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\).

The first option is equivalent to \(0^\sharp\)'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an extendible cardinal, either:

• Every regular cardinal greater than \(\delta\) is measurable in \(\mathrm{HOD}\).
• For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).

However, there is no known sharp for HOD that would cause the first option to hold.

An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is uncountable regular and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an extendible cardinal, then the following are equivalent:

• \(\mathrm{HOD}\) is a weak extender model for \(\delta\)'s supercompactness.
• There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD.

Then we have a strong dichotomy: if \(\delta\) is an extendible cardinal, either:

• Every regular cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD.
• No regular cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD.

None of these three statements are particularly hard to prove. The HOD hypothesis says that there is a proper class of cardinals \(\lambda\) which are not \(\omega\)-strongly measurable in HOD: therefore, if there is an extendible cardinal and the HOD hypothesis holds, then \(\mathrm{HOD}\) is close to \(V\). Currently, it's not known if the successor of a singular strong limit cardinal of uncountable cofinality can ever be \(\omega\)-strongly measurable in HOD - therefore, there is reason to believe in the HOD hypothesis. Interestingly, the HOD hypothesis implies the following. Letting \(T = \mathrm{Th}_{\Sigma_2}(V)\) be the \(\Sigma_2\)-theory of \(V\) with ordinal parameters (note that the notation \(\mathrm{Th}_{\Sigma_2}(V)\) typically indicates allowance arbitrary set parameters), i.e. \(T\) contains all the true formulae which are ZF-provably equivalent to a formula of the form \(\exists \alpha (V_\alpha \models \psi(\beta_1, \beta_2, \cdots, \beta_n)\) for an arbitrary formula \(\psi\) and ordinals \(\beta_1, \beta_2, \cdots, \beta_n\). Then there is no nontrivial elementary embedding \(j: (\mathrm{HOD}, T) \to (\mathrm{HOD}, T)\).

The HOD conjecture is the assertion that the theory of ZFC plus "there is a supercompact cardinal" proves the HOD hypothesis. Unlike the HOD hypothesis itself, this is an arithmetic statement. The HOD conjecture has some remarkable consequences, which don't require the axiom of choice: one of the most surprising is a possibility to eliminate the usage of choice within the proof of Kunen's inconsistency. Namely, if the HOD conjecture is true and \(\delta\) is extendible, then, for all \(\lambda > \delta\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\), and this doesn't require the axiom of choice.