Distinguishing Variants of Friedman's Property |
Hannes Jakob |
Dienstag, 7. Januar 2025 14:30Uhr |
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We introduce parametrized variants of Friedman's property. $F(\lambda,\kappa)$ states that any function from $\kappa$ into $\lambda$ is constant on a closed set of order type $\omega_1$. The principle $F^+((D_i: i\in\omega_1),\kappa)$ (for $(D_i : i\in\omega_1)$ a partition of $\omega_1$) states that for any sequence $(A_i: i\in\omega_1)$ of stationary subsets of $E_{\omega}^{\kappa}$ there is a normal function $f\colon\omega_1\to\kappa$ such that $f[D_i]\subseteq A_i$. We will prove all possible implications between instances of both properties and show the optimality of our results by obtaining suitable independence results. |