4
$\begingroup$

Separation principle $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ claims that any two disjoint pointsets of boldface class $\mathbf\Sigma^1_n$ are separated by a $\mathbf\Delta^1_n$ set. Separation principle $\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$ is understood similarly. It is known classically that $\mathbf{Sep}(\mathbf\Sigma^1_1,\mathbf\Delta^1_1)$ and $\mathbf{Sep}(\mathbf\Pi^1_2,\mathbf\Delta^1_2)$ hold, while $\mathbf{Sep}(\mathbf\Pi^1_1,\mathbf\Delta^1_1)$ and $\mathbf{Sep}(\mathbf\Sigma^1_2,\mathbf\Delta^1_2)$ fail. The axiom $V=L$ (anyway, the existence of a good $\mathbf\Delta^1_2$ wellordering of the reals) implies that $\mathbf{Sep}(\mathbf\Pi^1_n,\mathbf\Delta^1_n)$ holds, while $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ fails for all $n\ge3$. The axiom of projective determinacy $\mathbf{PD}$ implies some other behaviour. But very little seems to be known about $\mathbf{Sep}$ in various generic models, including such best known ones like adding $\kappa$-many Cohen reals, $\kappa\ge\omega_1$. Basically, afaik this is 1) Harrington's unpublished handwritings of 1974-75, and my own APAL, 2016, 167, 3, 262–283. I wonder can someone share some other results in this direction.

Kanovei, Vladimir; Lyubetsky, Vassily, Counterexamples to countable-section $\varPi_2^1$ uniformization and $\varPi_3^1$ separation, Ann. Pure Appl. Logic 167, No. 3, 262-283 (2016) ZBL06529281 MR3437647

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.