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
His proof was erroneous, but easily corrected; that
in 1951 by Los and Ryll-Nardzewski
and presented in detail in 1972 by
Plastria. The error involved this intermediate
(K) Any product of cofinite topologies is compact.
(with J. Alan Alewine)
Topologizing the Denjoy Space by Measuring Equiintegrability.
Real Analysis Exchange
31 (2005-06), 23-44.
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
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
stronger than both
and the topology given by
the Alexiewicz seminorm,
but it lacks the countability
and compatibility conditions that
are often associated with inductive
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
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.
|LOGIC||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).|
|(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.|
|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.]|
(with Daniel Biles)
Solvability of a finite or infinite system of discontinuous quasimonotone
Proceedings of the American Mathematical
128 (2000), 3349-3360.
This paper proves the existence of solutions to the initial value problem
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.
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
|ALGEBRA||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.|
|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.|
|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.|
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|