Large Cardinals and Their Inner Models (Math 143, Spring

Large Cardinals and Their Inner Models
(Math 143, Spring 2014)
Peter Koellner
2 Arrow Street, Room 414
[email protected]
This course is an introduction to large cardinal axioms (also known as axioms of infinity). These axioms extend the axioms of ZFC by asserting the
existence of higher and higher levels of infinity.
The mathematical study of the infinite is a subject that is of interest in
its own right. But, in addition, the theory of large cardinals has important
applications. Let me mention two. First, large cardinals play a central role in
measuring the consistency strength of strong mathematical theories. Second,
large cardinal axioms settle many of the questions that are independent of
ZFC. Thus the subject is important both from a mathematical and from a
philosophical point of view.
In the first part of the course we will scale the hierarchy of large cardinals,
starting, in our initial ascent, with the small large cardinals—inaccessible
cardinals, Mahlo cardinals, and indescribable cardinals—and then, we will
reach our first significant base camp—measurable cardinals—and our journey will begin in earnest, passing up through strong cardinals, Woodin cardinals, supercompact cardinals, extendible cardinals, huge cardinals, and
the large cardinals at the level of L(Vλ+1 ) and far beyond into the upper
reaches of the higher infinite.
In the second part of the course we will take stock and try to obtain an
deeper understanding of the hierarchy that we ascended in the first part.
During the initial ascent we will have noticed that all of the large cardinals
we reached could live in the constructible universe, L, which means that
in this universe (which is like a fine-tuned lens) we could see the details of
the large cardinals at a very high resolution. But upon reaching our first
significant base camp—measurable cardinals—we reached a large cardinal
that cannot exist in L (by a theorem of Scott). This raises the question of
whether there is an “L-like” model—a model with a detailed “fine-structure”
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that will reveal the details of all in sight—which can contain a measurable
cardinal (and thereby reveal its detailed structure). The program of inner
model theory is the program of finding such “L-like models” for larger and
larger large cardinals. This program has had great success. For example,
Dodd and Jensen found and “L-like” model that can accomodate a measurable cardinal. Unfortunately, this model cannot accomodate significantly
larger large cardinals, like strong cardinals. Strong cardinals are to the
Dodd-Jensen model what measurable cardinals are to L—they completely
transcend the model. This pattern continued. It turned out that for every large cardinal that could be reached by inner model theory there was
a larger large cardinal that “struck it down”. For this reason it looked like
inner model theory would be a long, endless march up the large cardinal
hierarchy, where we would have to create an entirely new inner model for
each transition point of the large cardinal hierarchy. And since there are an
endless number of transition points in the large cardinal hierarchy the attempt to understand large cardinals through their inner models looked like
a hopeless one. However, recently Woodin made an important discovery.
He showed that if one can solve the inner model problem for one supercompact cardinal then the inner model “goes all the way”, that is, there is an
“overflow” and the model for one supercompact cardinal is, in fact, able
to accomodate not just the supercompact cardinal but all large cardinals
(that live in the ambient universe). In the second part of the course we will
introduce the rudiments of this theory and describe one of its outstanding
open problems—the HOD Conjecture.
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Reading
The main reference for the first part of the course is my set of notes entitled
Very Large Cardinals. The main reference for the second part of the course
will be sections 4,5, and 7 of Hugh Woodin’s paper Suitable Extender Models,
I.
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Meetings
I will lecture each week on Wednesday from 1–3. The lectures will be rather
concise. We will schedule an additional hour that will be run like a workshop.
The purpose of the section will be to (a) cover some additional background
(if this proves necessary), (b) go through some of the proofs in more detail,
and (c) discuss some of the associated philosophical issues.
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Prerequisites
This is a rather advanced course in set theory and so there are some prerequisites.
I will assume basic set theory (ordinals, cardinals, cumulative hierarchy,
transfinite induction) and I would like to assume a bit more (club and stationary sets and the constructable hierarchy). The former is a necessary
prerequisite for taking the course. The latter is not strictly necessary as we
shall be going over it (but rather quickly) during the first few sections.
For the former, Chapters 1-3 of my “Set Theory” (on the course webpage) suffice for this (for more details see Enderton’s “Introduction to Set
Theory”)). For the latter, pages 91-96 of Jech’s “Set Theory: The Third
Millennium Edition” suffices for club and stationary sets and chapter 6 of
my “Set Theory” suffices for constructibility.
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Requirements
The final grade will be based on section participation (15%) and problem
sets (the remaining 85%).
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