A distillation of seminar notes for the seminars up through February 22, with emphasis on the last one ("Himalayan Expedition, Phase 1"--see abstract below) can be found here in pdf form and here in ps form and here in dvi form.

All seminars are at 3:30 in LeConte 312 unless otherwise noted.

The last seminar of the semester was:

Date: April 19
Speakers: Sandy Johnson and Mark Walters
Title: Cleanup with Sandy and Mark

During the semester, quite a lot of interesting facts were stated without proof due to lack of time. This final seminar of the semester featured proofs of a few basic facts used in proving Ulam's Dichotomy:

THEOREM [Ulam] Assume the axiom that there is a (countably additive) nonatomic measure on some set X such that every subset of X is measurable and some subset of X has positive real measure. Then EXACTLY ONE of the following holds:
(1) There is a subset Y of positive real measure m such that every subset of Y has measure either 0 or m, and
(2) There is a subset Y of positive measure and having no more members than there are real numbers; and there is a measure extending Lebesgue measure on the real line such that every subset is measurable.

Seminar 1
Date: January 25, 2006
Speaker: Prof. Peter Nyikos
Title: Interactions between set theory and measure theory

Abstract: The concept of a countably infinite set is basic to much of modern mathematics, including analysis. Cantor's revolutionary discovery that the real line is an uncountable set will be shown using the ordinary (Lebesgue) measure on the real line. Then we will explore applications of set theory to measure theory, including the existence of nonmeasurable subsets of the real line and various drastic efforts to overcome this ``deficiency.'' In particular, the replacement of countable additivity of measure by finite additivity runs afoul of the Banach-Tarski paradox in 3-dimensional Euclidean space. A highly readable treatment of this paradox can be found in a 1980 master's thesis done here at USC by Dan Toth.

Seminar 2
Date: February 1
Title: How many real numbers are there? (Part 1)
Speaker: Prof. Peter Nyikos

Abstract: Modern set theory (and also much of modern mathematics) can be said to have started with Cantor's revolutionary discovery that there is more than one infinity; specifically, that there are more real numbers than there are integers, even though there are infinitely many integers. "More" here is stronger than saying "in addition to." For example, Cantor also showed that there are just as many integers as there are rational numbers altogether, and just as many points on a line segment as there are in 3-dimensional Euclidean space.

Cantor also tried to answer the question of which infinite number expresses the cardinality of the real line. This seminar talk surveys the various possibilities, with special emphasis on what Cantor thought was the right answer: that this number is the smallest infinite number greater than the one for the integers. This claim, called the Continuum Hypothesis, is now known to be independent of the usual axioms of set theory, as will be explained (though not proven!) in this talk.

Seminar 3
Date: February 8
Title: How many real numbers are there? (Part 2)
Speaker: Prof. Peter Nyikos

Abstract: Cantor developed the theory of infinite ordinal and cardinal numbers, which will be explained in this seminar, along with well-ordered sets in general.
Cantor tried to answer the question of which infinite number expresses the cardinality of the real line. We continue to survey the various possibilities, with special emphasis on what Cantor thought was the right answer: that this number is the smallest infinite number greater than the one for the integers. This claim, called the Continuum Hypothesis, is now known to be independent of the usual axioms of set theory. This is explained using models of set theory, which might be called "island universes" (the old term for galaxies). They are sets that mimic the class of all sets so closely that they satisfy every property which is generally believed to be satisfied by the class of all sets.

Seminar 4
Date: February 15
Speaker: Prof. Peter Nyikos
Title: Interactions between set theory and measure theory, Part 2

We return to the theme of the first seminar talk but also pull together some themes from the second and third talks, in which talked a bit about well-ordered sets and cardinal numbers, and looked at the question "How many real numbers are there?" Cantor's Continuum Hypothesis (CH) states that the number describing it is the second infinite cardinal number. Cantor launched modern set theory by showing that it is not the first infinite cardinal number, but was never able to prove that it is the second. And with good reason: we now know that all the generally accepted axioms and techniques of mathematics are inadequate to settle this question.

While recalling what we need from the first three lectures, we take another look at well-ordered sets and look at some consequences of CH for measure theory. These include a result so remarkable that Godel counted it as evidence against CH being "really true" despite having himself shown that CH cannot be disproved using the accepted axioms and techniques of mathematics. We also address the question of whether it is consistent that there is a probability measure on [0, 1] with respect to which every subset of [0, 1] is measurable and every finite subset has measure 0. We do know that CH implies this is false, but that is just the beginning!

Seminar 5
Date: February 22
Speaker: Prof. Peter Nyikos
Title: Himalayan Expedition, Phase 1

We embark on an expedition to what might be called the loftiest mountain range in mathematics, the range of large cardinal numbers. We set our sights on one of the highest ridges in the range, the ridge of measurable cardinals. In this first phase we make our way to our base camp, nestled in the first ridge, the ridge of inaccessible cardinals, while catching glimpses from time to time of the ridge of measurable cardinals. Perhaps there will also be time to explore a little beyond our base camp before the seminar is over.

Part of the motivation for this expedition comes from some axioms introduced in the preceding seminar lecture. One is equivalent to the statement that there is a (countably additive) nonatomic measure on some set X such that every subset of X is measurable and some subset of X has positive real measure. The key dichotomy concerning this axiom is:
(1) There is a subset Y of positive finite measure m such that every subset of Y has measure either 0 or m, and
(2) There is no such Y.
It is (1) that characterizes the first measurable cardinal; it is enormously greater than the cardinality of the real line. In his doctoral dissertation, Stanislaw Ulam showed that (2) is equivalent to there being an extension of Lebesgue measure to a measure where every subset of the real line is measurable. A remarkable discovery of Solovay is that (1) is consistent if one assumes the consistency of (2), and vice versa.

This first of two phases is self-contained: what is needed from earlier lectures in this seminar series will be provided. It should also be easy for any graduate student in this department to understand. The second phase will be more demanding.

Seminar 6
Date: March 1
Speaker: Prof. Peter Nyikos
Title: Himalayan Expedition, Phase 2

We take off from the base camp we set up at the very end of Phase 1 of the expedition, climbing past inaccessible, hyper-inaccessible, and Mahlo cardinals, to get a better glimpse of where the first uncountable measurable cardinal must be in the high Himalayas of set theory, and to get a better look also at how enormous the set of all real numbers must be if what I call the second alternative of Ulam's Dichotomy holds:

THEOREM [Ulam] Assume the axiom that there is a (countably additive) nonatomic measure on some set X such that every subset of X is measurable and some subset of X has positive real measure. Then EXACTLY ONE of the following holds:
(1) There is a subset Y of positive real measure m such that every subset of Y has measure either 0 or m, and
(2) There is a subset Y of positive measure and having no more members than there are real numbers; and there is a measure extending Lebesgue measure on the real line such that every subset is measurable.

Seminar 7
Date: March 15
Speaker: Kate Scott
Title: Godel's first incompleteness theorem

Kate Scott makes her debut as a seminar speaker in a presentation outlining the proof of Godel's first incompleteness theorem. Godel's discovery in 1930 of this theorem had a profound impact not only in mathematics but also in philosophy, by showing that knowledge is open-ended: there will always be some mathematical statements that are true but cannot be shown true by any definable set of consistent axioms. In fact, such statements can be found in elementary number theory, and we only need number theory (and a good dose of mathematical logic!) to show this theorem.

Seminar 8
Date: March 29
Speaker: Prof. Peter Nyikos
Title: Lines and trees

Linearly (i.e., totally) ordered sets and what set theorists call trees are special cases of partial orders. The real line, the rational line, and the set of natural numbers each have simple characterizations up to an order-preserving bijection. The classical characterizations for the rationals and reals will be given, and the latter leads naturally into the concept of a Souslin line, whose existence is independent of the usual axioms of set theory and which is theoretically important in measure theory. The intimate connection between Souslin lines and Souslin trees will be explained. No prior attendance at the seminar is assumed and the lecture will be on a level understandable to all graduate students.

Seminar 9
Date: April 5
Speaker: Prof. Peter Nyikos
Title: Ladders and orderings

A ladder is an sequence that climbs to the top of an infinite set without a greatest member, so to speak. In 1903, G. H. Hardy used a ladder system to produce an uncountable well-orderable set of real numbers. We will go over Hardy's proof (with modern terminology) and show a couple of other uses of ladder systems. No prior attendance at the seminar is assumed and the lecture will be on a level understandable to all graduate students.

Seminar 10
Date: April 12
Speaker: Prof. Peter Nyikos
Title: Himalayan Expedition, Phase 3

In the first two phases of the expedition, we saw how hopeless it was to reach the first uncountable measurable cardinal by climbing up from below. In this third phase, it will be as though we were picked up by the wind and blown though dense clouds and deposited on a high slope, with no idea of how far we have been blown and how far we have to go. Many cardinal numbers reached in this way have been obtained in analogy with finite Ramsey theory, which produces extremely large finite numbers by defining numbers ``internally'' instead of ``from below.''

We will also be blown far beyond the first uncountable measurable cardinal to a high peak from which we can deduce surprising things about much smaller sets. Call a family A of countable subsets of the real line uncluttered if for each A in A there are only countably many sets produced by intersecting A with all other members of A . Does there exist an uncluttered collection A such that every countable subset of the real line is a subset of some member of A ? This question has fascinating implications for very large infinite cardinal numbers.

No prior attendance at the seminar is assumed and the lecture will be on a level understandable to all graduate students.