9
$\begingroup$

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?

$\endgroup$
11
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.