The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals
Abstract
We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, squarekappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.
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