HOD dichotomy: Difference between revisions
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* No [[Cofinality|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 [[ZFC|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|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.
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