Title: Nyikos's Joke
Author: Arnold W. Miller
Date: March 2016
Abstract:
In 1988 Peter Nyikos asked me if there were any Q-sets in Laver's
model. After I solved it (Theorem 16), I asked Peter why he wanted to
know. He said he was just joking, ``no Q-points / no Q-sets'',
referring to my paper 1980 paper: There are no Q-Points in Laver's
Model for the Borel Conjecture.
At the time I was more interested in Theorem 12 which says that for any
countable ordinal \alpha, Borel set B a subset of
(\omega^\omega)^\alpha, and any condition p is the P_\alpha (iteration
of Laver forcing) there is a condition q\leq p such that [q] is a
subset of B or is disjoint from B.
Paul Larson recently asked me (May 2015) if I knew how to prove there
are no Q-sets in the iterated perfect set model. Theorem 12 and 16 are
true for iterated Sacks forcing, as well as, many other tree forcings,
e.g. Superperfect forcing, Mathias forcing, Silver forcing, or mixed
versions.
Latest version:
http://www.math.wisc.edu/~miller/res/ny.pdf