Schedule:
Talks will be at Ryerson 251. Lunch and coffee at the Barn
(Ryerson 352). Door will be open on Eckhart Hall
Friday.
Lunch at noon, at the Barn (Ryerson Hall 352).
1:15 - 2:00 --
Stephen
G. Simpson.
Some
aspects of reverse mathematics
2:15 - 3:00 --
Steffen Lempp.
Reverse mathematics and
first-order arithmetic
Coffee break
3:30 - 4:15 --
Carl
Jockusch.
Ramsey's
Theorem and arithmetic comprehension
4:30 - 5:15 --
Jiayi
Liu.
Cone
avoid closed sets induced by non-enumerable trees within
partition
Saturday.
Breakfast at 9:30, at the Barn (Ryerson Hall 352)
9:55 - 10:10 --
Presentation of the
Reverse
Math Zoo by
Damir
Dzhafarov
10:15 - 11:00 --
Jeff
Hirst.
Two
combinatorial proofs and some related questions
11:15 - 12:00 --
Carl
Mummert.
Reverse
mathematics of principles equivalent to the axiom of choice
Lunch
2:15 - 3:00 --
Michael
Rathjen.
Omega
models and well-ordering principles
3:15 - 4:00 --
Jeremy
Avigad.
Computability,
constructivity, and convergence in measure theory
Coffee Break
4:30 - 5:15 --
Andreas Weiermann.
Well partial orders and reverse
mathematics
5:30 - 6:15 --
Alberto
Marcone.
The
strength of 'interval wqos are bqos'
Sunday.
Breakfast at 9:00, at the Barn (Ryerson Hall 352)
9:15 - 10:00 --
Peter
Cholak.
Some projects in Reverse
Mathematics
10:15 - 11:00 --
Theodore A. Slaman.
Infinite Random Sequences and
First Order Consequences
11:15 - 12:00 --
Richard
A. Shore.
Weak
Principles and Low Levels of Induction
Lunch
Abstracts:
Jeremy Avigad
Title:
Computability,
constructivity, and convergence in measure theory
Abstact: When confronted with a nonconstructive theorem, those
interested in computational and constructive aspects of
mathematics have various options: they can seek constructive and
computational versions of the theorem in question; they can
calibrate the extent to which the given theorem it
nonconstructive; and they can look for computational or other
explicit information hidden in the proof. I will discuss
measure-theoretic convergence theorems that come up in the study
of dynamical systems and stochastic processes, and give examples
of all three approaches to understanding their constructive
content.
Peter Cholak
Title:
Some projects in
Reverse Mathematics
Abstract: We will discuss some recent work in reverse
mathematics. We will discuss a recent paper of Cholak,
Galvin, and Solomon relating Ramsey's theorem, infinite
traceable graphs and finitely generated infinite lattices, an
ongoing project of Cholak and Dzhafarov, Schweber, Shore on c.e.
and co-c.e. partial orders and work of Cholak's student, Flood,
that will appear in Flood's Ph.D. thesis, hopefully in
2012. If time permits other ongoing projects might be
discussed.
Jeff Hirst
Title:
Two combinatorial
proofs and some related questions
Abstract: We will examine the proof that Hindman's theorem
implies arithmetical
comprehension and the proof that Ramsey's theorem for pairs
implies Friedman's
free set theorem for pairs with the goal of revealing a common
theme. The talk
will conclude with a list of questions about related
combinatorial theorems.
Carl Jockusch
Title:
Ramsey's
Theorem and arithmetic comprehension
Abstract: This will be an elementary expository talk on
the strength of various forms of Ramsey's Theorem in comparison
with the system ACA_0. In particular, it will include a
description of a simplified proof (due to Damir Dzhafarov and
me) of David Seetapun's 1995 result that Ramsey's theorem for
pairs does not imply ACA_0 in the base system RCA_0.
Steffen Lempp
Title: Reverse mathematics and
first-order arithmetic
Abstract: I will start by giving a brief introduction to
the work of Paris and others from the 1970's about theories of
first-order arithmetic. I will then connect this to more
recent work in reverse mathematics, focusing on the
first-order consequences of second-order theories, in
particular those arising from Combintorics.
Jiayi Liu
Title:
Cone avoid closed sets
induced by non-enumerable trees within partition
Abstract: The relation between combinatorial property and
computational property of “an object” (a real, a set of real or
so) has recently draw some attention. We sketch a proof of one
of such theorem that, for a given \Pi^1_0 binary tree T, and a
set C, if C does not compute a strong enumeration of T (in a
non-trivial way), then for every set A, there exists an infinite
subset G of either A or its complement, such that the joint
degree of C and G also does not compute any strong enumeration
of T. We give applications of this result including, RT_2^2 does
not imply WWKL_0.
Alberto Marcone
Title:
The strength of
'interval wqos are bqos'
Abstract: In 2006 Pouzet and Sauer proved the following theorem:
if an interval order is wqo, then it is bqo. We present some
preliminary results about the reverse mathematics of this
theorem, showing connections with some of the open problems in
this area, such as the Generalized Higman's Theorem and '3 is
bqo'.
Carl Mummert
Title:
Reverse
mathematics of principles equivalent to the axiom of choice
Abstract: We study the reverse mathematics of certain maximality
principles that are known to be equivalent to the axiom of
choice when formalized in $\mathsf{ZFC}$. These principles
have a surprising range of strengths, including examples that
are equivalent to full second-order arithmetic and an example
that is weaker than $\mathsf{ACA}_0$ and incomparable with
$\mathsf{WKL}_0$. Our work also illustrates the inner
combinatorics of the principles, which can be obscured in the
context of set theory. This is joint work with Damir Dzhafarov.
Michael Rathjen
Title:
Omega models and
well-ordering principles
Abstract. The purpose of this talk is to present a general
methodology which in many cases allows one to establish an
equivalence between two types of statements. The first type is
concerned with the existence of omega models of a theory whereas
the second type asserts that a certain (usually well-known)
elementary operation on orderings preserves the property of
being well-ordered. These results were inspired by work of
Friedman, Marcone and Montalban. The primordial example is
Friedman's characterization of the theory ATR_0 by means of a
Pi-1-2 sentence of the form "if X is well ordered then f(X) is
well ordered", where f is a standard proof theoretic function
from ordinals to ordinals. The approach taken here, however, is
rather different in that the methods used are purely
proof-theoretic and crucially involve cut elimination theorems
in infinitary logic with ordinal bounds.
One could perhaps generalize and say that every cut
elimination theorem in ordinal-theoretic proof theory
encapsulates a theorem of this type. It is also interesting to
ponder the question whether the technique has the potential for
generalization, namely whether it can be lifted up to
beta-models and functors acting on ordinal functions.
Richard A. Shore
Title:
Weak Principles and
Low Levels of Induction
Abstract: Our usual approach in talks on reverse
mathematics is to simplify the situation by taking the
computational point of view and dealing only with standard
models. In this talk, we will instead try to present some of the
delicate issues that arise when one is faced with proofs or
constructions that (seem to) need more induction than the
standard base theory (RCA_0) provides. These issues arise both
when one chooses (among classically equivalent versions of)
basic definitions and again when one does proofs and
constructions. We will discuss the types of problems that arise
and various ways of dealing with them: choosing the "right"
definitions and theorems, giving "better" proofs or showing that
more induction is "really needed". We will draw our examples
primarily from recent work with Hirschfeldt and Lange on
homogenous models and some earlier work with Hirschfeldt and
Slaman on atomic models.
Stephen Simpson
Title:
Some aspects of
reverse mathematics
Abstract: In this talk we comment on some philosophical aspects
of reverse mathematics. Among the philosophical issues
considered are finitism, potential infinity, actual infinity,
predicativity, and predicative reductionism. In addition,
we take this opportunity to announce some new results concerning
algorithmic randomness, Kolmogorov complexity, and the reverse
mathematics of measure theory.
Theodore A. Slaman
Title:
Infinite Random
Sequences and First Order Consequences
Abstract: We will discuss the question of what first-order
or number-theoretic consequences can be drawn from the second
order hypothesis that for every real X there is another Y such
that Y is random relative to X.
Andreas Weiermann
Title: Well partial orders and reverse mathematics
Abstract: Harvey Friedman showed that ATR_0 does not prove
Kruskal's theorem. Following this line of research we discuss
the role of well partial orders in reverse mathematics with a
particular emphasis on recent developments.