March 31
Corey Switzer,
University of Vienna
Galois-Tukey reductions and canonical structure in the Cichoń diagram
Cardinal invariants of the continuum are cardinal numbers which, roughly, measure how 'badly' CH fails in various mathematical contexts such as analysis and topology. For instance the cardinal ${\rm add}(\mathcal N)$ is the least $\kappa$ for which there are $\kappa$ many Lebesgue measure zero sets of reals whose union is not measure zero. Classical facts imply $\aleph_1 \leq {\rm add}(\mathcal N) \leq 2^{\aleph_0}$ but the precise value is undetermined in ZFC and depends heavily on the axioms of set theory. Other numbers follow a similar pattern of 'the least size of a set of reals (Borel sets, etc) lacking a classical smallness property'.
The Cichoń diagram displays cardinal invariants related to Lebesgue measure (the null ideal), Baire category (the meager ideal) as well as the bounding and dominating numbers which concern growth rates of functions. Many surprising ZFC-inequalities exist between these cardinals suggesting a rich world living on the reals in various models of set theory. At the combinatorial heart of every proof of a ZFC inequality derives from a Galois-Tukey reduction: the (ZFC-provable) existence of a pair of continuous maps with simple properties that make sense outside of the context of logic and indeed would be sensible to any analyst or topologist.
In this talk we will discuss some recent work in progress on the descriptive complexity of maps witnessing consistent but non-provable implications. We will show using largely computability theoretic methods that in Gödel's constructible universe there are low level projective reductions between any two cardinal invariants - thus CH holds in a very 'definable' way, while in Solovay's model of 'all sets of reals are Lebesgue measurable' (and therefore the axiom of choice fails) there are no non-ZFC provable implications thus these cardinals are all as different as possible.