HEXAGONbeta

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

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

Contributed by Elliot Glazer

SubmitterElliot Glazer

Version 4 / 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

References

  1. S. Roguski, The theory of the class HOD , in A. Lachlan, M. Srebrny, and A. Zarach (eds.), Set Theory and Hierarchy Theory V , Lecture Notes in Mathematics 619 , Springer-Verlag, Berlin, Heidelberg, 1977, pp. 251--255. https://doi.org/10.1007/BFb0067656 doi:10.1007/BFb0067656 .DOI
  2. Core-model rank reduction and removal of Foundation , prior research notes, 23 September 2026, 05:13--06:07 UTC. Preserved in the supplied research archive and reproduced in the accompanying archive excerpt.
  3. T. Benhamou, Prikry forcing and tree Prikry forcing of various filters , https://arxiv.org/abs/1801.04424 arXiv:1801.04424 , version 2.arXiv
  4. S. Grigorieff, Intermediate submodels and generic extensions in set theory , Annals of Mathematics (2) 101 (1975), no. 3, 447--490.
  5. M. Gitik, Y. Hayut, and A. Karagila, The first measurable can be the first inaccessible cardinal , https://arxiv.org/abs/2401.02757 arXiv:2401.02757 , version 2.arXiv
  6. F. Schlutzenberg, The definability of the extender sequence $ E$ from $ E _1$ in $L[ E]$ , https://arxiv.org/abs/1906.00276 arXiv:1906.00276 .arXiv
  7. J. D. Hamkins, Gap forcing: generalizing the L\'evy--Solovay theorem , https://arxiv.org/abs/math/9901108 arXiv:math/9901108 .arXiv
  8. T. Usuba, Geology of symmetric grounds , https://arxiv.org/abs/1912.10246 arXiv:1912.10246 , version 3.arXiv
  9. W. J. Mitchell, The Covering Lemma , in Handbook of Set Theory , Springer, 2010. https://math.ufl.edu/ wjm/files/covering.pdf Author's draft , Sections 1 and 3--4.link
  10. W. J. Mitchell, The complexity of the core model , https://arxiv.org/abs/math/9210202 arXiv:math/9210202 .arXiv
  11. R. Schindler and J. R. Steel, The core model induction , https://math.berkeley.edu/ steel/papers/corebookjul07.pdf July 2007 draft .link
  12. R. Schindler, Coding into $K$ by reasonable forcing , https://arxiv.org/abs/math/9706208 arXiv:math/9706208 .arXiv
  13. J. R. Steel and S. Zoble, Determinacy from strong reflection , https://math.berkeley.edu/ steel/papers/mmcnov2012.pdf November 2012 draft .link
  14. J. R. Steel, An Introduction to Iterated Ultrapowers , https://www.math.cmu.edu/users/jcumming/Appalachian/steel_cmu_2015_files/Iteration-Ultrapowers.pdf Lecture notes , Lectures 1 and 5.link
  15. Y. Hayut and A. Karagila, Small measurable cardinals , https://arxiv.org/abs/2212.07973 arXiv:2212.07973 , version 2.arXiv
  16. S.-D. Friedman, Trapping cofinality , https://www.logic.univie.ac.at/ dsyfriedman/papers/trapping.cof.pdf December 2021 draft .link
  17. F. Schlutzenberg, HOD in inner models with Woodin cardinals , https://arxiv.org/abs/2012.07185 arXiv:2012.07185 .arXiv
  18. G. Goldberg, The Ultrapower Axiom , https://math.berkeley.edu/ goldberg/Papers/UABook.pdf Author's manuscript , Theorem 6.2.7.link
  19. H. Jakob, Slender trees and the approximation property , preprint, https://arxiv.org/abs/2401.13446 arXiv:2401.13446 , version 2. The iteration-like criterion is Theorem 3.8 in the HTML version consulted September 23, 2026.arXiv
  20. J. Reitz, Cohen forcing and inner models , preprint, https://arxiv.org/abs/1809.10092 arXiv:1809.10092 , version 2.arXiv
  21. Y. Cheng, S.-D. Friedman, and J. D. Hamkins, Large cardinals need not be large in HOD , https://arxiv.org/abs/1407.6335 arXiv:1407.6335 , Appendix, Lemmas 18--20.arXiv
  22. J. Reitz, The Ground Axiom , preprint, https://arxiv.org/abs/math/0609064 arXiv:math/0609064 .arXiv
  23. T. Usuba, The downward directed grounds hypothesis and very large cardinals , preprint, https://arxiv.org/abs/1707.05132 arXiv:1707.05132 .arXiv
  24. M. Carl, P. Schlicht, and P. Welch, Recognizable sets and Woodin cardinals: computation beyond the constructible universe , Annals of Pure and Applied Logic 169 (2018), no. 4, 312--332. https://doi.org/10.1016/j.apal.2017.12.007 doi:10.1016/j.apal.2017.12.007 .DOI
  25. W. J. Mitchell, The sharp for the Chang model is small , Archive for Mathematical Logic 56 (2017), 935--982. https://arxiv.org/pdf/1705.00208 Final arXiv manuscript .link
  26. T. Gappo and G. Sargsyan, Determinacy in the Chang model , Proceedings of the American Mathematical Society 154 (2026), 1763--1769. https://arxiv.org/pdf/2302.06487 arXiv manuscript .link
  27. A. Suzuki, No elementary embedding from $V$ into $V$ is definable from parameters , Journal of Symbolic Logic 64 (1999), 1591--1594. https://doi.org/10.2307/2586799 doi:10.2307/2586799 .DOI
  28. A. Karagila, Preserving dependent choice , Bulletin of the Polish Academy of Sciences. Mathematics 67 (2019), 19--29. https://doi.org/10.4064/ba8169-12-2018 doi:10.4064/ba8169-12-2018 .DOI
  29. W. J. Mitchell, Ramsey cardinals and constructibility , J. Symbolic Logic 44 (1979), no. 2, 260--266. https://doi.org/10.2307/2273732 doi:10.2307/2273732 .DOI
  30. A. W. Apter and P. Koepke, Making all cardinals almost Ramsey , Arch. Math. Logic 47 (2008), 769--783. https://doi.org/10.1007/s00153-008-0107-1 doi:10.1007/s00153-008-0107-1 . https://www.math.uni-bonn.de/people/koepke/Preprints/Making_all_cardinals_almost_Ramsey.pdf Author's version .DOI
  31. A. Kanamori and T. Awerbuch-Friedlander, The compleat $0^ $ , Math. Logic Quart. 36 (1990), no. 2, 133--141. https://doi.org/10.1002/malq.19900360206 doi:10.1002/malq.19900360206 .DOI
  32. A. Karagila, Fodor's lemma can fail everywhere , Acta Math. Hungar. 154 (2018), no. 1, 231--242. https://arxiv.org/abs/1610.03985 arXiv:1610.03985 .arXiv
  33. V. Gitman and T. A. Johnstone, Indestructibility properties of Ramsey and Ramsey-like cardinals , Ann. Pure Appl. Logic 173 (2022), no. 6, 103106. https://doi.org/10.1016/j.apal.2022.103106 doi:10.1016/j.apal.2022.103106 . https://victoriagitman.github.io/files/indestructibleramseycardinalsnew.pdf Author's version .DOI
  34. O. Ben-Neria and P. L\"ucke, On $ _1$-definable closed unbounded sets , Canad. J. Math. 78 (2026), no. 3, 769--801. https://doi.org/10.4153/S0008414X24000567 doi:10.4153/S0008414X24000567 .DOI

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