Working in ZF, it's well-known that for any $n \ge 2,$ the claim that there is a $\Sigma_n$ well-ordering of the universe is equivalent to the axiom $V=HOD.$ It seems natural to believe there should be a similar theorem for $n=1,$ perhaps that there being a $\Sigma_1$ well-ordering of the universe is equivalent to $V=L.$ I can't find any counterexamples to this proposition, having checked various generic extensions of $L$ and canonical inner models like $L[U],$ so I'm guessing this is true.

It would suffice to check that if there a $\Sigma_1$ well-ordering of $V,$ then all sets of ordinals are in $L.$ This is true for subsets of $\alpha$ for any $\alpha$ countable in $V,$ by applying Mansfield's theorem that if there is a $\Sigma_2^1$ well-ordering of $\mathbb{R},$ then $\mathbb{R} \subset L.$ Then the first non-trivial case would be to show this holds for subsets of $\omega_1,$ but I haven't been able to do this (the proof of Mansfield's theorem doesn't seem to extend to this more general case).

So, are there any known results similar to what I'm asking?


1 Answer 1


No, see the paper On $ Σ_1$ Well-Orderings of the Universe by Harrington and Jech.


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