Distinguishing Variants of Friedman's Property
Dienstag, 7.1.25, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
We introduce parametrized variants of Friedman's property. \(F(\blambda,\bkappa)\) states that any function from \(\bkappa\) into \(\blambda\) is constant on a closed set of order type \(\bomega_1\). The principle \(F^+((D_i: i\bin\bomega_1),\bkappa)\) (for \((D_i : i\bin\bomega_1)\) a partition of \(\bomega_1\)) states that for any sequence \((A_i: i\bin\bomega_1)\) of stationary subsets of \(E_{\bomega}^{\bkappa}\) there is a normal function \(f\bcolon\bomega_1\bto\bkappa\) such that \(f[D_i]\bsubseteq A_i\). We will prove all possible implications between instances of both properties and show the optimality of our results by obtaining suitable independence results.
Derivation-like theories and neostability
Dienstag, 28.1.25, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Motivated by structural properties of differential field extensions, Omar Leon Sanchez and I introduced the notion of a theory T being derivation-like with respect to another model complete theory \(T_0\). We proved that when T admits a model companion \(T^+\), then several model-theoretic properties are transferred from \(T_0\) to \(T^+\). These properties include completeness, quantifier elimination, stability, simplicity, and NSOP\(_1\). Examples of derivation-like theories are plentiful but are typically obtained by adding extra structure to theories of fields. In this talk I will introduce the central notions, detail how the proofs work by lifting independence relations from \(T_0\) to \(T^+\), and give examples.