{"id":"hexagon/2609.00114","pdf":"https://hexagonmath.org/pdf/2609.00114v5","url":"https://hexagonmath.org/abs/2609.00114v5","tags":["msc:03E99","Set theory","axiom of choice"],"score":0,"title":"Under Measurables, HOD_{Ord^{\\omega}} is locally arbitrary","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Elliot Glazer"],"reviews":0,"subject":"math.LO","summary":"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 of $\\ZFC$. For any sentence $\\sigma$, 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}…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex"],"discussed":0,"published":"2026-09-28","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}