User talk:RhubarbJayde/REL-NPR

Subdivision candidates

Some ideas for conditions for refining the hierarchy of ordinals

• \(\Sigma_1\land\Pi_1\) conditions were mentioned on GS, the idea coming from determinacy. \(\Sigma_1\land\Pi_1\) formula include the \(\Sigma_1\) formulas by \(\Sigma_1\land\top\) and the \(\Pi_1\) formulas by \(\top\land\Pi_1\), and every \(\Sigma_1\land\Pi_1\) formula is itself \(\Delta_2\).
• An Ershov hierarchy but for \(\Delta_2(L_\alpha)\) sets. The usual Ershov hierarchy has \(\mathcal D_n\) consist of the sets which are symmetric differences of \(n\) recursively enumerable sets, maybe this can be done for \(\Sigma_1(L_\alpha)\) sets (e.g. "\(\mathcal D_2\) but for \(L_\alpha\)" will be the sets definable by \(\Sigma_1\oplus\Sigma_1\) formulas on \(L_\alpha\).) The Ershov hierarchy has the benefit of transfinite extensions existing, for example here ("On a difference hierarchy for arithmetical sets") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets.
• \(\Sigma_1(St)\) formulas.
• To go below admissibility, the \(\{f\mid f\textrm{ is a restriction of }(\gamma,\delta)\mapsto\gamma+\delta\}\)-cardinals are the additively indecomposable \(\alpha\), this isn't very good but maybe something can come from the primitive recursive ordinal functions.

There may have to be an \(\alpha>0\) requirement throughout this post. C7X (talk) 23:21, 22 October 2023 (UTC)