Preface
The history of mathematical logic abounds with connections to other areas of
mathematics as well as other fields like philosophy. Set theory began when Can-
tor needed transfinite processes for his analysis of trigonometric series. The basic
notions of computability theory are needed in the study of topics ranging from
Hilbert's tenth problem (in number theory) to the word problem for groups (in
algebra but motivated by topology). Model theory is the foundation of modern
infinitesimal analysis. Proof theory elucidates fundamental questions of epistemol-
ogy.
Like most fields of mathematics, logic has, in addition to its external connec-
tions, a rich internal development, motivated by questions arising from within the
field itself. Examples include forcing, large cardinals, and inner models in set the-
ory; stability in model theory; functional interpretations and ordinal analysis in
proof theory; and the priority method in computability theory. It seems fair to
say that, during the last half of the twentieth century, research in mathematical
logic was concerned more with such internal issues than with outreach to the rest
of mathematics.
Recently, however, outreach has become stronger. The proof theorists' func-
tional interpretations have developed to the point where they can give explicit
estimates in analysis, sometimes better than what was obtained by analytic meth-
ods. Descriptive set theory, especially the branch that deals with the complexity of
equivalence relations, has established connections with fields ranging from abelian
group theory to ergodic theory. Technical constructions in model theory have been
shown to have fascinating connections with Schanuel's conjecture in transcendental
number theory. Computably enumerable sets have infiltrated Riemannian geome-
try.
The present volume arose from two back-to-hack conferences held at the Univer-
sity of Michigan in April, 2003. The first, on "Logic and Its Applications in Algebra
and Geometry," sought to bring together logicians (particularly model theorists and
set theorists) working in these areas. The second, a workshop on "Combinatorial
Set Theory, Excellent Classes, and Schanuel Conjecture," put more emphasis on
pure logic, though, as suggested by the presence of Schanuel's conjecture in the
title, connections with other areas were present as well.
Yi Zhang, who works on set-theoretic questions in group theory, spent the
academic year 2002-2003 in my department at the University of Michigan. Shortly
after he arrived, he asked me whether he could organize a conference here. I pointed
out that we had no money for that. My comment turned out to be irrelevant; he
volunteered to organize a conference anyway. I'm not sure how one gets prominent
logicians to travel to a conference when no financial support is available, but Yi
v
Previous Page Next Page