## Seminar in Set Theory and Logic, Fall 2006

The abstracts for the Spring semester seminar talks may be found here

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

NOTE CHANGED TIME

Time and location: Wednesday, December 6, 12:20 pm in LeConte 312

Speaker: Peter Nyikos

Title: Almost disjoint families and trees

NOTE: This seminar serves as a make-up for a class that was cut short by Carrie Finch's MS presentation, but it is on a topic that will be understandable to anyone able to follow the abstract below. It was originally scheduled for the usual seminar time but was replaced by McNulty's fourth and final lecture.

ABSTRACT: A family of infinite sets is said to be almost disjoint if the number of elements in the intersection of any two of them is less than the number in either one. For instance, if every member of the family is denumerable, then any two of them intersect in a finite set.

Trees will be used to show some results about almost disjoint families, particularly maximal ones. Every maximal infinite almost disjoint family of subsets of the integers is uncountable, but the number may depend on axioms beyond the usual ZFC axioms of set theory. Where families of uncountable sets are involved, even the maximum size of an almost disjoint family is axiom-sensitive.

If time permits, trees will be used to outline a proof of one part of a theorem due to Erdos and Rado in partition calculus.

#### Monday, December 4

[Note: this is a change from what was posted earlier] Time and location: 3:40 - 4:40 pm in LeConte 312

Speaker: George McNulty

Title: Collapsing to One Equation

We will prove that every finite set of equations which includes the axioms for rings with unit is logically equivalent to a single equation. This is the third and final step needed to establish that there is no algorithm which determines upon input of an equation whether it is compatible with the real line.

#### Monday, November 27

Time and location: 3:40 pm in LeConte 312

Speaker: George McNulty

Title: A topological algebra over the reals with an undecidable equational theory

The equational theory of (R, +, *, -,1,| |, sin*) is just the set of all equatons true in this algebraic system. We will prove that there is no algorithm which, upon input of an equation, will eventually answer the question of whether the equation is true in our algebraic system. This turns out to be a consequence of the negative solution to Hilbert's Tenth Problem. It is also the second of three steps toward the problem of deciding which equations are compatible with the real line.

This is the seminar meeting postponed from Monday, November 20.

#### Monday, October 23

Speaker: Peter Nyikos

Title: Rings of functions and ultrafilters and nonstandard analysis

This is a self-contained continuation of last week's seminar. Those who missed last week's seminar can find the material covered here in ps form and here in pdf form

#### Monday, October 16

Time and location: 3:40 pm in LeConte 312

Speaker: Peter Nyikos

Title: Rings of functions and ultrafilters

Abstract: For each set X, the set of all real-valued functions on X is a ring under addition and multiplication. Maximal ideals on this ring naturally correspond to ultrafilters on X.

The quotient field modulo a maximal ideal naturally contains the field of real numbers, and the question of whether this containment is proper is equivalent to that of whether there is a countable subset of the ultrafilter whose intersection is empty. This in turn gets us into talk of measurable cardinals.

When the containment is proper, the field can be used as the foundation for nonstandard analysis, a 1960's discovery of Abraham Robinson which finally put the calculus of Newton and Leibnitz on a firm foundation by giving a meaning to their "fluxions" and "infinitesimals".

This talk was half survey, half proofs accessible to anyone who knows the basics of ring theory. Ultrafilters and measurable cardinals are defined and explained "from scratch."

#### Monday, October 9

Time and location: 3:40 pm in LeConte 312

Speaker: Peter Nyikos

Abstract: A ladder at   x   is a strictly ascending sequence whose supremum is   x. Ladder systems on the well-ordered set omega_1 are used to prove Theorems 1 and 3 in the first abstract below.

An easy construction will be given of an Aronszajn tree using Theorem 3. No attendance at previous seminars is assumed.

All one needs to know about omega_1 is that it is an uncountable set that is well-ordered in such a way that each element has only countably many predecessors.

An Aronszajn tree is an uncountable tree in which every chain and every level is countable.
A tree is a partially ordered set (poset for short) in which the predecessors of every element are well ordered.
A chain in a poset is a totally ordered subset.
The levels of a tree are indexed by ordinal numbers, and the elements on level a are the ones whose set of predecessors has order type a.

The second seminar of the Fall semester was on Monday, September 25.

Time and location: 3:40 pm in LeConte 312

Title: Some infinite combinatorial principles and an application, Part II

Speaker: Prof. Peter Nyikos

Abstract: Three theorems about coherent systems were discussed and a proof was given of Theorem 2:

Theorem 1: The axiom "club" implies that there is a coherent 2-coloring of the countable subsets of omega_1 which is not uniformizable on any uncountable subset of omega_1.

[All one needs to know about omega_1 is that it is an uncountable set that is well-ordered in such a way that each element has only countably many predecessors.]

Theorem 2: The axiom CC_11 implies that every coherent 2-coloring of the countable subsets of omega_1 is uniformizable on some uncountable subset of omega_1.

Theorem 3: There is a coherent 2-coloring of the countable subsets of omega_1 which is not uniformizable on all of omega_1.

Axiom CC_11 has to do with ideals on omega_1. A collection I of subsets of a set S is called an ideal if
(1) The union of any two members of I is in I and
(2) if J is a subset of a member of I then J is also in I.

An ideal is countable-covering if for all countable sets Q, there is a countable subcollection J of I such that the intersection of every member of I with Q is a subset of some member of J.

Axiom CC_11 says that for every countable-covering ideal I on omega_1, at least one of the following is true:
(1) there is an uncountable subset S of omega_1 such that every countable subset of S is a member of I and
(2) there is an uncountable subset X of omega_1 which meets every member of I in a finite set.

Definitions of "coherent" and "uniformizable" can be found below in the abstract for the first seminar.

The first seminar of the semester in set theory and logic was Monday, September 18.

Time and location: 3:40 pm in LeConte 312

Title: Some infinite combinatorial principles and an application

Speaker: Prof. Peter Nyikos Abstract: There are several very strong principles of infinite combinatorics that are independent of the usual axioms of set theory. This talk will focus on two of the simplest, "club" and CC_11, with very different effects, each implying the other is false. Their contrasting properties will be demonstrated in talking about coherent systems on the minimal uncountable well-ordered set W.

A coherent system on W is a collection of 2-colorings of all the countable subsets of W, such that the 2-colorings of any two countable subsets agree in all but finitely many elements. Given an uncountable subset Z of W, a coherent system is said to be uniformizable on Z if there is a 2-coloring of Z which agrees with every 2-coloring in the coherent system in all but finitely many places.

It is possible, just using the ordinary ZFC axioms of set theory, to construct a coherent system on W which is not uniformizable on W itself. However, CC_11 implies that each coherent system is uniformizable on some uncountable subset of W. In contrast, the principle "club" allows one to construct a coherent system which is not uniformizable on any uncountable set.