Appalachian set theory
September 9, 2006
Carnegie Mellon University
9:30 a.m. to 6 p.m. (with breaks) in Baker Hall A53
(Click for campus map)
Paul Larson
:
"An introduction to Pmax forcing"
The partial order Pmax, invented by W. Hugh Woodin, is
one a family of forcings which produce models of ZFC when applied to
models of determinacy. The partial order is given by a directed
system of models of ZFC, and grew out of Woodin's proof that the
existence of a measurable cardinal plus the saturation of the
nonstationary ideal on ω1 implies the failure of the
Continuum Hypothesis. The power set of ω1 in the
Pmax extension is maximal in that it satisfies all
Pi2 sentences whose forceability is implied by large
cardinals. In this way the structure P(&omega1) in
the Pmax extension is very similar to the one given by
Martin's Maximum. Advantages of the Pmax approach
include reduced large cardinal hypotheses (at the level of
determinacy hypothesis) and avoidance of iterated forcing
technology. Variations of Pmax are very useful for consistency
results on P(ω1); some produce models in which
the Boolean algebra P(ω1)/NSω1 is
&omega1-dense. Larson and Todorcevic's consistency proof for
the statement that every compact space with T5 square is
metrizable grew out of another Pmax variation.
This one-day course will have two aims. The first is to start the
participants along the shortest path towards applying
Pmax in their own work. The second is to prepare them
for reading Woodin's book on Pmax, The axiom of
determinacy, forcing axioms and the nonstationary ideal, whose
technical prerequisites are fairly high. We will start with the
fundamental combinatorial arguments underlying Pmax,
and at some point shift gears to a higher-level approach. We will
black box the most technical issues in the basic analysis, the
existence of Pmax conditions, though the proofs will
be made available. Taking this step for granted, most of the
Pmax analysis can be carried out using techniques
which we should be able to present in full. The main reference will
be my expository article
Forcing over
models of determinacy,
written for the Handbook of Set Theory and available in
DVI.
and PDF.
Links:
Background reading
for this workshop
Post-workshop materials:
Lecture notes from the workshop (PDF; Revised 10/09)
List of participants in this workshop
