CUNY Logic WorkshopFriday, October 18, 20132:00 pmGC 6417

# Generic choice functions and ultrafilters on the integers

## Paul B. Larson

### Miami University of Ohio

We will discuss a question asked by Stefan Geschke, whether the existence of a selector for the equivalence relation $E_0$ implies the existence of a nonprincipal ultrafilter on the integers. We will present a negative solution which is undoubtedly more complicated than necessary, using a variation of Woodin’s $mathbb{P}_{mathrm{max}}$. This proof shows that, under suitable hypotheses, if $E$ is a universally Baire equivalence relation on the reals, with countable classes, then forcing over $L(E,mathbb{R})$ to add a selector for $E$ does not add a nonprincipal ultrafilter on the integers.

Professor Larson will also be speaking at the Rutgers MAMLS the following two days.

Professor Larson (B.S. Dartmouth, Ph.D. UC Berkeley) conducts research in set theory, particularly on the topic of forcing and large cardinals.

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on September 7th, 2013