Logic Seminar Abstracts


Title: Satisfiability of linear equations on [0,1].
Speaker:  Walter Taylor
Affiliation:  CU
Time:  3pm, Monday, January 26
Location:  Math 220

Abstract:  
We consider the satisfiability of a set $ \Sigma$ of equations on a space $ A$, i.e. the existence of continuous $ A$-operations that satisfy the equations $ \Sigma$. In my Kempner Colloquium talk on 12/1/03, I outlined a proof that there is no algorithm to decide if a finite $ \Sigma$ is satisfiable on $ \mathbb{R}$ .

In the seminar, we shall restrict our attention to linear $ \Sigma$. (Here an equation is called linear if each of its terms contains either no function symbols or a single function symbol.) In this context we prove that if $ \Sigma$ is consistent (i.e. it has a model of more than one element), then $ \Sigma$ is satisfiable on [0,1] and on $ \mathbb{R}$ . (We shall prove the result for [0,1] in detail, while offering only a sketch for $\mathbb{R}$.)

How much logic will one need to follow the proof? Very little: at a certain point in the proof I will use a few linear equations that have been seen to lie in $ \Sigma$ and deduce the inconsistent equation x = y, thereby finishing off a proof by contradiction. And how much topology? Not a whole lot: we will study a metric on a quotient space that is defined by a closed equivalence relation, and to the resulting space we will apply Tietze's Extension Theorem.


Title: Satisfiability of linear equations on [0,1].
Speaker:  Taylor
Affiliation:  CU
Time:  3pm, Monday, February 2
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title: Satisfiability of linear equations on [0,1].
Speaker:  Taylor
Affiliation:  CU
Time:  3pm, Monday, February 9
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Countable choice and excluded middle.
Speaker:  Fred Richman
Affiliation:  Florida Atlantic University
Time:  3pm, Monday, February 16
Location:  Math 220

Abstract:  
A mostly expository talk touching on The Weyl-Polya bet, The Myhill-Goodman proof that full choice implies excluded middle, Arguments for and against countable choice, Examples of life without countable choice (and excluded middle): Cauchy reals versus reals, The Hilbert syzygy theorem, Trace-class operators, The fundamental theorem of algebra.


Title:   Recently discovered properties of the group of order preserving permutations of the real line.
Speaker:  Charles Holland
Affiliation:  Bowling Green State University
Time:  3pm, Monday, February 23
Location:  Math 220

Abstract:  
A lattice-ordered group is both a group and a lattice in which the lattice operations are respected by all group translations. The group Aut(R) of all order preserving permutations of the real line R is a lattice-ordered group when given the pointwise order, and it is universal in the sense that every countable lattice-ordered group embeds into it. The group Aut(R), therefore, has been much studied by those interested in lattice-ordered groups. Only recently has it been discovered that Aut(R) has some remarkable "cofinality" properties. I will discuss a theorem of which the following two are corollaries:

1. Aut(R) is not the union of a countable tower of proper subgroups.

2. If S is any set of generators of Aut(R) then there is a positive integer n such that every member of Aut(R) is a product of no more than n elements of S and their inverses.


Title:   Recently discovered properties of the group of order preserving permutations of the real line.
Speaker:  Charles Holland
Affiliation:  Bowling Green State University
Time:  3pm, Monday, March 1
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Which T0  topological algebras are Hausdorff?
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, March 8
Location:  Math 220

Abstract:  
The variety of groups has the property that every T0  topological algebra in the variety is Hausdorff ( = T2). In this talk we discuss what is known about the characterization of the class of varieties for which the implication "T0 implies T2" is valid.


Title:   Which T0  topological algebras are Hausdorff?
Speaker:  Keith Kearnes
Affiliation:  CU
Time:  3pm, Monday, March 15
Location:  Math 220

Abstract:  
A continuation of the previous talk.


Title:   Modeling equations on $ {\mathrm{\mathbb{R}}}$ with $ C^n$ operations.
Speaker:  Walter Taylor
Affiliation:  CU
Time:  3pm, Monday, April 5
Location:  Math 220

Abstract:  
It is an ongoing project to consider the compatibility of a set $ \Sigma$ of equations with a space $ A$, i.e. the existence of continuous $ A$-operations that satisfy the equations $ \Sigma$. In particular, we may ask the question for $ A = {\mathrm{\mathbb{R}}}$. In my Kempner Colloquium talk on 12/1/03, I outlined a proof that there is no algorithm to decide if a finite $ \Sigma$ is satisfiable on $ {\mathrm{\mathbb{R}}}$, and in my seminar talks of 1/26, 2/2, 2/9/04, I proved compatibility of those $ \Sigma$ that are linear and consistent.

Now we change the topic only slightly, and consider satisfiability of $ \Sigma$ on $ {\mathrm{\mathbb{R}}}$ using operations of class $ C^n$ (i.e. they have continuous derivatives of order $ n$). In the seminars of 4/5 and 4/12/04, we shall prove, for each $ n$, that there exists $ \Sigma_n$ that is satisfiable on $ {\mathrm{\mathbb{R}}}$ using $ C^{n-1}$-operations, but not using $ C^n$-operations. (This means, incidentally, that for $ m\neq n$, the clone of $ C^m$ operations is not elementarily equivalent to the clone of $ C^n$ operations.)

The relevant $ \Sigma_n$ is a finite set of equations that forces the use of a non-trivial spline $ f_n$ of order $ n$. More precisely, $ \Sigma_n$ forces $ f_n$ to have compact support, to be piecewise a polynomial of degree $ n$, and to be non-trivial. Therefore, obviously, it cannot be of class $ C^n$.

The equations $ \Sigma_n$ that force the use of $ f_n$ include: the axioms of commutative ring theory; some ancillary equations; and most notably, a so-called scaling equation for $ f_n$.

These talks have few prerequisites beyond Math 2400. Splines will be introduced from scratch - it's a big theory, but the part we need is small and easily explained. The over-arching project here is certainly that of a general-algebraic or logical understanding of [the category of] topological spaces, but the April lectures may be followed without any background in logic or general algebra.


Title:   Modeling equations on $ {\mathrm{\mathbb{R}}}$ with $ C^n$ operations.
Speaker:  Walter Taylor
Affiliation:  CU
Time:  3pm, Monday, April 12
Location:  Math 220

Abstract:  
A continuation of the previous talk.



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