Completeness of Ordered Fields and a Trio of Classical Series Tests

This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of R. The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field.

THE CLASSICAL CONTINUUM WITHOUT POINTS

We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary “actual infinity”. Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishop’s (1967) constructivism, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a nonpunctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind–Cantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

Infinitesimals in ordered vector spaces

An infinitesimal approach to ordered spaces is proposed. Archimedean property and Dedekind completeness in ordered spaces are discussed from a nonstandard point of view.

Constructing Banaschewski compactification without Dedekind completeness axiom

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications ofX, namely, the Stone-Čech compactificationβX, the Banaschewski compactificationβ0X, and the structure space𝔐X,Fof the lattice-ordered commutative ringℭ(X,F)of all continuous functions onXtaking values in the ordered fieldF, equipped with its order topology. Some open problems are also stated.

Dedekind Completeness and the Algebraic Complexity of o-Minimal Structures

An ordered structure is o-minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o-minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary.) We prove that if a theory is not binary then it is not rc-ary for any n. We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T, and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T, then there is an interval I in so that if is a Dedekind complete elementary extension of .

Dedekind Completeness and a Fixed-Point Theorem

McShane (5, 6) has introduced the concept of “Dedekind completeness” for partially ordered sets, which seems to be a natural generalization of the usual concept of completeness for lattices. It is the purpose of this paper to discuss some of the properties of Dedekind completeness, particularly with respect to a rather natural class of partially ordered sets which we call “uniform.”