Sherwood Hachtman : Borel determinacy, level by level: monotone induction and reflection principle

Posted by Dima Sinapova , part of the Logic Seminar.

At

April 21, 2015, 4 p.m.

In

SEO 427

Abstract

Borel determinacy, though a theorem of ordinary analysis, cannot be proven without some appeal to the higher infinities of set theory: By the dual results of Harvey Friedman and Donald Martin, the strength of $\Sigma^0_{1+\alpha+3}$-determinacy is roughly that of ZF with Power set restricted to $\alpha+1$ iterations on $\omega$. Refining these results, we eliminate the "roughly", isolating a family of novel reflection principles whose strengths correspond exactly to that of determinacy for these levels. We will describe this work, also mentioning connections with higher-order reverse mathematics, and stronger determinacy principles having the strength of measurable cardinals. We also discuss some recent work building on that of Philip Welch, giving a new, natural characterization of the strength of $\Sigma^0_3$-determinacy in terms of monotone operators.