User talk:RhubarbJayde/REL-NPR: Difference between revisions

* \(\Sigma_1\land\Pi_1\) 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 [https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/on-transfinite-levels-of-the-ershov-hierarchy/96FCFB2D3989682C95CA1387364A6B58 transfinite extensions existing], for example here ("[https://homepages.ecs.vuw.ac.nz/~melnikal/sigma03beyond.pdf On a difference hierarchy for arithmetical sets]") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets.

* 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 [https://mathoverflow.net/q/388619 primitive recursive ordinal functions].