August 7
Brent Cody,
Virginia Commonwealth University
Higher indescribability
To what extent can formulas from infinitary logics be used in set-theoretic reflection arguments? If $\kappa$ is a measurable cardinal, any $L_{\kappa,\kappa}$ sentence which is true in $(\kappa,\in)$, must be true about some strictly smaller cardinal. Whereas, there are $L_{\kappa^+,\kappa^+}$ sentences of length $\kappa$ which are true in $(\kappa,\in)$ and which are not true about any smaller cardinal. However, if $\kappa$ is a measurable cardinal and some $L_{\kappa^+,\kappa^+}$ sentence $\varphi$ is true in $(\kappa,\in)$, then there must be some strictly smaller cardinal $\alpha<\kappa$ such that a canonically restricted version of $\varphi$ holds about $\alpha$. Building on work of Bagaria and Sharpe-Welch, we use canonical restriction of formulas to define notions of $\Pi^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$. In this context we show that such higher indescribability hypotheses are strictly weaker than measurability, we prove the existence of universal $\Pi^1_\xi$-formulas, study the associated normal ideals and notions of $\xi$-clubs and prove a hierarchy result. Time permitting we will discuss some applications.