### 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 P_{max} forcing"

The partial order P_{max}, 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 ω_{1} implies the failure of the
Continuum Hypothesis. The power set of ω_{1} in the
P_{max} extension is maximal in that it satisfies all
Pi_{2} sentences whose forceability is implied by large
cardinals. In this way the structure P(&omega_{1}) in
the P_{max} extension is very similar to the one given by
Martin's Maximum. Advantages of the P_{max} approach
include reduced large cardinal hypotheses (at the level of
determinacy hypothesis) and avoidance of iterated forcing
technology. Variations of P_{max} are very useful for consistency
results on P(ω_{1}); some produce models in which
the Boolean algebra P(ω_{1})/NS_{ω1} is
&omega_{1}-dense. Larson and Todorcevic's consistency proof for
the statement that every compact space with T_{5} square is
metrizable grew out of another P_{max} variation.
This one-day course will have two aims. The first is to start the
participants along the shortest path towards applying
P_{max} in their own work. The second is to prepare them
for reading Woodin's book on P_{max}, **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 P_{max},
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 P_{max} conditions, though the proofs will
be made available. Taking this step for granted, most of the
P_{max} 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