Calliope Ryan-Smith (University of Leeds)

Large cardinals for independent families

A collection of subsets of the natural numbers is independent if all finite intersections of members of the collection with all finite intersections of complements of members of the intersection is never empty. These are interesting combinatorial families and it is easy enough to prove their existence. Furthermore, by Zorn's Lemma, one can always extend an independent family to be maximal (that is, contained in no further independent family). However, as soon as one tweaks these parameters, specifically the word 'finite', existence becomes much harder to work with. Kunen showed that even having a maximal $\sigma$-independent family (so we take countable intersections) implies that there is an inner model with a measurable cardinal, so this is already not provable from ZFC. Indeed, he also showed that this goes the other way: Beginning with a measurable cardinal, there is a forcing extension in which $2^{\omega_1}$ has a maximal $\sigma$-independent family. Moving up the ladder, one obtains from an $\aleph_1$-strongly compact cardinal a forcing extension in which there are maximal $\sigma$-independent families for a class of cardinals. The talk shall briefly introduce the basics of large cardinals and go over this method of converting large cardinals into maximal independent families.