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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

SubmitterElliot Glazer

Version 3 / 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 V, HOD is an arbitrary model ZFC model. For any sentence \sigma, letting CM denote "class of measurables," the following theories are equiconsistent: (1) ZFC + CM + [HOD_{Ord^\omega} \models (\exists \kappa V_{\kappa} \models \sigma)]; (2) ZFC + GCH + CM + [HOD_{Ord^\omega} \models (SVC \wedge CM \wedge \exists \kappa V_{\kappa} \models \sigma)]; (3) ZF + DC + CM + \exists \kappa V_{\kappa} \models \sigma. Thus, under measurables we have that HOD_{Ord^{\omega}} is "locally arbitrary."

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. I have not yet worked through all of Astra's coding details myself.

Tools used

OpenAI
Codex

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Version history

  1. v1Initial depositSupersededSep 28, 2026
  2. v2Submitted as a new versionSupersededSep 28, 2026
  3. v3Submitted as a new versionSupersededSep 29, 2026
  4. v4Submitted as a new versionCurrentSep 30, 2026