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Under Measurables, HOD_{Ord^{\omega}} is locally arbitrary

Primarily AI-generated textHuman understanding: some partsmath.LO — Logic

Contributed by Elliot Glazer

Version 5 / Sep 28, 2026 / CC BY 4.0

Abstract

For the inner model $\HOD_{\Ord^{\omega}}$, we derive an analogue for Roguski's classical result that without nontrivial assumptions on VV, $\HOD$ is an arbitrary model of $\ZFC$. For any sentence σ\sigma, letting $\CM$ denote ``class of measurables,'' the following theories are equiconsistent: \begin{enumerate} \item $\ZFC+\CM+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CM+ [\HOD_{\Ord^{\omega}}\models (\SVC\wedge\CM\wedge\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZF+\DC+\CM+ \exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate} Thus, under measurables the only local constraint on $\HOD_{\Ord^{\omega}}$ is the requisite choice principle. We also derive an analogue for Woodins. Letting $\CW$ denote ``class of Woodins,'' the following are equiconsistent: \begin{enumerate} \item $\ZFC+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]+$ $\HOD_{\Ord^{\omega}}$ is a ground of VV; \item $\ZFC+\CW+$ some symmetric extension $V(G) \models \DC+\exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate}

Provenance statement

I sketched out the strategy for proving the main results of this paper to Astra in Codex. I had not ironed out the finer details of the Prikry coding needed to make the proof work so Astra played a substantial role in completing this paper. Partial formalization has been achieved, in particular of the novel Prikry coding details. Full formalization will require a substantial inner model theory library.

Tools used

OpenAI
CodexVersion 6 Astra

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Version history

  1. v1Submitted by Elliot GlazerInitial depositSupersededSep 28, 2026
  2. v2Submitted by Elliot GlazerSubmitted as a new versionSupersededSep 28, 2026
  3. v3Submitted by Elliot GlazerSubmitted as a new versionSupersededSep 29, 2026
  4. v4Submitted by Elliot GlazerSubmitted as a new versionSupersededSep 30, 2026
  5. v5Submitted by Elliot GlazerSubmitted as a new versionCurrentOct 12, 2026