SET
THEORY
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Kelley's specialization of Tychonov's Theorem is
equivalent to the Boolean Prime
Ideal Theorem, Fundamenta Mathematicae
189 (2006), 285-288. In 1950 John L. Kelley published
the first proof that Tychonov's Theorem (TT) implies the Axiom of
Choice (AC).
His proof was erroneous, but easily corrected; that
was mentioned
in 1951 by Los and Ryll-Nardzewski
and presented in detail in 1972 by
Plastria. The error involved this intermediate
principle:
(K) Any product of cofinite topologies is compact.
The implication
TT => K is trivial, but
Kelley's proof of K => AC
was faulty. The 1951 and 1972 papers
replaced K with other, more
complicated principles, thus
leaving open this interesting
question: is
K => AC actually
true by some other proof, or is it false?
My paper shows that it's false.
In fact, K turns out to be equivalent
to BPI, which is well known to be strictly weaker than AC.
145kb pdf
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REAL
ANALYSIS
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(with J. Alan Alewine)
Topologizing the Denjoy Space by Measuring Equiintegrability.
Real Analysis Exchange
31 (2005-06), 23-44.
Basic limit
theorems for the KH integral involve
equiintegrable sets. We construct a family
of Banach spaces
whose bounded sets are precisely the
subsets of KH[0,1] that are
equiintegrable
and pointwise bounded. Then
KH[0,1] is the union of these
Banach spaces, and can be topologized
as their inductive limit.
That topology is
is barreled,
bornological, and
stronger than both
pointwise convergence
and the topology given by
the Alexiewicz seminorm,
but it lacks the countability
and compatibility conditions that
are often associated with inductive
limits.
239kb pdf
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LOGIC
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Classical and Nonclassical Logics. 507 + ix pages.
Princeton University Press, August 2005.
(This is not research, but rather a textbook -- i.e. it is not new discoveries, but my attempt to make
more understandable some ideas that were already present in the research literature.)
So-called classical logic is just one of the many kinds of reasoning present in everyday thought. When presented by itself -- as in most introductory texts on logic -- it seems arbitrary and unnatural to students new to the subject.
CNL introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics;
the contrast illuminates all of them. Two of P.U.P.'s internal reviewers said
It will be a valuable addition to the literature, especially as some of the topics covered have long been the preserve of specialists.
The exposition is solid and successfully clarifies topics that traditionally are difficult to understand by a novice. ... The author shows that he has great ability to lucidly describe complicated ideas in various schools of logic.
Math Reviews (MR2164393) said
Let us note that this is an introductory textbook for which no previous experience with mathematical logic is required. The author's style is clear and approachable, and, consequently, this book seems to be ideal for beginning students of both mathematics and philosophy, as well as for students of computer science and the large circle of logicians working in the field of nonclassical logics.
For further description see
the book's web page, which includes a downloadable excerpt.
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LOGIC
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Equivalents of Mingle and Positive Paradox. Studia
Logica
77 (2004), 117-128.
Abstract:
Relevant logic is a proper subset of classical logic. It does not include among
its theorems any of
positive paradox, mingle,
linear order, or
unrelated extremes.
This article shows that those four formulas have different effects when added to relevant
logic, and then lists many formulas that have the same effect as positive paradox or mingle.
Preprint downloadable here
(105 kb PDF file).
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REAL
ANALYSIS
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(with J. Alan Alewine) Review of the recent book
The Integral: An Easy Approach after Kurzweil and
Henstock,
by Peng Yee Lee and Rudolf Vyborny. Reviewed in
the
American
Mathematical Monthly
108 (2001), 577-582. The
review discusses this book and (to a lesser extent)
some other, related books, with particular attention to
where these books belong in our standard curriculum.
You can
read this paper
online
or download
preprints available in several formats.
For a slightly different introduction to the KH integral,
see my
web page on the subject.
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PHILOSOPHY
OF MATH | |
Constructivism is difficult.
American Mathematical Monthly 108
(2001), 50-54. Abstract:
Constructivism is unusually difficult to learn.
Learning most mathematical subjects merely involves adding a little to
one's knowledge, without disturbing what one already has, but learning
constructivism involves
modifying all aspects of what one already knows:
theorems, methods of reasoning, technical
vocabulary, and even the use of everyday words that do not seem
technical, such as "or". In this paper I discuss, in the language of
mainstream mathematicians, some of those modifications; perhaps
newcomers to constructivism will not be so overwhelmed by it if
they know what kinds of difficulties to expect.
[Paper can be viewed online in
html
format. For better looking printed copies, please
instead download the
pdf version.]
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DIFFERENTIAL
EQUATIONS
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(with Daniel Biles)
Solvability of a finite or infinite system of discontinuous quasimonotone
differential equations.
Proceedings of the American Mathematical
Society
128 (2000), 3349-3360.
Abstract:
This paper proves the existence of solutions to the initial value problem
x'(t)=f(t,x(t)) (0£t£1), x(0)=0
where
f : [0,1]´RM ® RM
may be discontinuous but is assumed to satisfy conditions of
superposition-measurability, quasimonotonicity, quasisemicontinuity,
and integrability. The set M can be
arbitrarily large (finite or infinite); our theorem is new even for
card(M)=2. The proof is based
partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by
Rzymowski and Walachowski. Further generalizations are
mentioned at the end of the paper.
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ANALYSIS
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Handbook of Analysis and its Foundations,
published by Academic Press. (Hardback, 1996/1997; 883 + xxii pages long.
The CD-ROM published in 1999 contained an additional chapter.)
(This is not research, but an expository/reference work. That is, instead of new discoveries, it
attempts to make more accessible to students some results that were already present
in the literature.) The book's main themes are -- and indeed, its prepublication title was -- Choice, Compactness, Completeness; those are three
main methods of proving the existence of a mathematical object. HAF explores
how these arise in analysis and in fields closely connected with analysis.
The exploration
of the Axiom of Choice is based on a brief, intuitive discussion of constructivism, which
S.I.A.M. Review called "the most satisfying reflection on constructivism I have ever seen"; in general that review called the book
"daring and innovative." The book
shows the connections underlying many different parts of mathematics that are usually
presented separately. The book also explores the limitations of
many basic principles of analysis -- for instance, every analyst knows Banach's
Contraction Fixed Point Theorem, but few know the converses of Meyers and
Bessaga, which show that in certain respects Banach's result cannot be improved.
The book includes some of the "folklore" results -- i.e., basic
ideas that are well known to advanced researchers but that graduate students sometimes
have difficulty locating in the literature. S.I.A.M. Review said
Every once in a while a book comes along that so effectively redefines an educational enterprise -- in this case, graduate mathematical training -- and so effectively reexamines the hegemony of ideas prevailing in a discipline -- in this case, mathematical analysis -- that it deserves our careful attention. This is such a book. There is nothing else remotely similar to it in any of the current books on integration, real analysis, set theory, or any other related subject. ...
The book is has a
web page of its own
which describes the book further and includes some brief excerpts -- e.g., charts and lists
of about 100 forms of the Axiom of Choice and its consequences.
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ALGEBRA
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Review of "Solving
the Quintic", a poster by Wolfram Research. My review of
Wolfram's poster was published in the Mathematical
Intelligencer 17 (1995), 71-73. Review is
suitable for beginners -- it includes an introduction
to the problem of solving polynomial equations by
formulas analogous to the quadratic formula.
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SET
THEORY
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Two topological equivalents of the Axiom of Choice. Zeit. fur math.
Logik und Grund. Math. 38 (1992), 555-557. We show that the Axiom of
Choice is equivalent to each of the following statements: a product of
closures of subsets of topological spaces is equal to the closure of
their product (in the product topology); and a product of complete
uniform spaces is complete.
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DIFFERENTIAL
EQUATIONS
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A survey of local existence theories for abstract nonlinear initial
value problems. Springer Lecture Notes in Math. 1394 (1989), 136-184.
This paper surveys the abstract theories concerning local-in-time
existence of solutions to differential inclusions, u'(t) in
F(t,u(t)), in a Banach space. Three main approaches assume
generalized compactness, isotonicity in an ordered Banach space, or
dissipativeness. We consider different notions of "solution," and
also the importance of assuming or not assuming that F(t,x) is
continuous in x. Other topics include Caratheodory conditions,
uniqueness, semigroups, semicontinuity, subtangential conditions, limit
solutions, continuous dependence of u on F, and bijections between
u and F.
Reprint available in several formats.
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DIFFERENTIAL
EQUATIONS
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Compact perturbations of linear m-dissipative operators which lack
Gihman's property. Springer Lecture Notes in Math. 1248 (1987),
142-161. Some questions about abstract methods for initial value
problems lead us to a study of the equation (*) u'(t) = (A+B)u(t),
where A is m-dissipative and B is compact. Does a solution to (*)
necessarily exist? Earlier studies of this question, reviewed and then
continued here, depend on an analysis of the related quasiautonomous
equation (**) u'(t) = Au(t)+f(t). We say A has Gihman's
property if the mapping f |® u is continuous from
L1w([0,T],K) into C([0,T];X) for
every compact set K contained in X; this condition is closely related to the
Lie-Trotter-Kato product formula. If A has this property, then (*)
is known to have a solution. In this paper, we consider linear,
m-dissipative operators A which lack Gihman's property. We obtain
partial results regarding the existence of solutions of (*); but in
general, the existence question remains open. Our method applies the
variation of parameters formula to (**), but this requires a weakened
topology when Range(f) is not contained in the closure
of Domain(A). Two
examples are studied: one in
l¥, the other in the space of
bounded continuous functions. |