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Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$?

Note that $Σ_2^V$ is the best possible since under the assumptions, existence of a non-ordinal definable real is (using replacement) a $Π_2^V$ statement true in $V$ but not in $\mathrm{HOD}$. Also, $Σ_2^{V,y}$ means $Σ_2^V$ using $y$ as a parameter.

An equivalent formulation is $|\mathrm{OD}∩ℝ|=ω ∧ ∀λ∈Ord \, ∃κ \, \mathrm{Theory}(\mathrm{HOD}∩V_κ,∈) = \mathrm{Theory}(V_λ,∈)$.

Variations: A stronger version (given $|\mathrm{OD}∩ℝ|=ω$) is existence of a $Σ_2$ elementary embedding of $\mathrm{HOD} ∩ V_λ$ into $V_λ$ where $λ$ is the least ordinal such that for every $y∈\mathrm{OD}∩ℝ$ every true $Σ_2^{V,y}$ statement holds in $V_λ$. (A variation is to use "$∃λ'$" and $\mathrm{HOD} ∩ V_{λ'}$ instead.) However, if not too strong, this version might be too difficult with the current techniques, so I used the weaker version above. One can also consider variations relative to a real parameter $x$ and $\mathrm{HOD}_x$. There are also many strengthenings of $|\mathrm{OD}∩ℝ|=ω$ such as determinacy of ordinal definable games on ordinals of length $ω_1$.

Motivation

The motivation for the question is that higher canonical models capture more and more truth: $\mathrm{HYP}$ can compute $Σ^1_1$ truth, $L$ is $Σ^1_2$ correct, $M_1$ can compute $Σ^1_3$ truth, $M_ω$ can compute $L(ℝ)$ truth, and so on. Among such models, a natural $(V,∈)$ definable limit appears to be $\mathrm{HOD}$; and in canonical models of determinacy, $\mathrm{HOD}$ is in a sense the core model.

A reasonable conjecture is that $\mathrm{HOD}$ is (in terms of truth) close to $V$ — and the question tests consistency of the conjecture (with the answer presumably using forcing to give an example without however having $\mathrm{HOD}$ as a canonical model). However, even if consistent, the conjecture might be false if for example measurable cardinals exist in $V$ and in $\mathrm{HOD}$, but behave differently in $V$ than in $\mathrm{HOD}$.

Thus, one view is that in place of $V$ $=$ "ultimate $L$", $\mathrm{HOD}$ acts as an ultimate $L$, with $\mathrm{HOD}$ being small in terms of the number of sets (and relationship to $V$), but large in terms of correctness.

However, I think that treating $\mathrm{HOD}$ as the ultimate $L$ is an overstatement as (under a reasonable platonic view not shared by everyone) using canonical indiscernibles for $V$ (also called $ω$-reflective cardinals), we can extend the language of set theory and define real numbers and canonical models beyond $\mathrm{HOD}$.

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  • $\begingroup$ What is HYP? The minimal model? $\endgroup$ Feb 11, 2019 at 12:14
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    $\begingroup$ @ElliotGlazer I think HYP is intended to be the model of hereditarily hyperarithmetical sets, also known as $L_\alpha$ where $\alpha=\omega_1^{CK}$ is the first non-recursive ordinal. $\endgroup$ Feb 11, 2019 at 13:18

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