Logic Seminar

Please note that the time is PST

A result due to Schrittesser and the speaker states:
Theorem 1: If ΓΓ is a somewhat "reasonable" class of subsets of Baire space which have the properties:
(1) All sets are completely Ramsey (i.e., Baire measurability w.r.t. the Ellentuck topology); and
(2) Uniformization (even just on Ramsey positive sets);
then there can't be any infinite maximal almost disjoint ("mad") families in ΓΓ.

The question arises if we can replace (1) by Laver measurability (in the sense of A. Miller), i.e.
(1') Every set in ΓΓ either contains the set of branches through a Laver tree, or it avoids all branches through a Hechler tree.

In this talk, I will show this is not the case by constructing a Π11Π11 infinite mad family in the Laver forcing extension of L (noting here that the class Π11Π11 has uniformization, and by a result of A. Miller (1') holds in the in the Laver forcing extension of L).

The reason this is interesting (to me, at least) is that this shows that despite a number of similarities between generic reals for Laver and Ramsey measurability (and Laver and Mathias forcing), Laver can't replace Ramsey in the proof of Theorem 1.

This is joint work with David Schrittesser.