Dedekind Cuts Research Articles

ORDERED STABILITY AND EXPANSIONS OF A PURE LINEAR ORDER BY A UNARY FUNCTION

A linearly ordered structure is said to be o-stable if each of its Dedekind cut has a “small” number of extensions to complete 1-types. This concept, which was introduced by B.S. Baizhanov and V.V. Verbovsky, generalizes such widely known concepts among specialists in model theory as weak o-minimality, (weak) quasi-o-minimality and dp-minimality of ordered structures. It is based on a combination of the concepts of o-minimality and stability. As we know, the elementary theory of any pure linear order is o-superstable. Indeed, this follows from the fact, which Rubin proved in the late 70s of the 20th century, that any type of one variable is determined by its cut and definable subsets, distinguished by unary predicates or formulas with one free variable. In this paper, we explore the question of what happens if a pure linear order is expanded with a unary function. Two examples were constructed when o-stability is violated; in addition, sufficient conditions for preserving o-stability with such language expansion were found. Research work on this topic is not yet finished, ideally, it would be good to find a criterion for preserving ordered stability when enriching a structure with pure linear order with a new function of one variable.

Continuity in Logic of Sense: Deleuze, Leibniz, Dedekind

ABSTRACT This essay explores the possibility of a metaphysical concept of continuity, which seems to have an implicit though decisive presence in Deleuze’s thought. It exposes a peculiar continuity that animates the indiscernibility of borders without making its constitutive elements homogenous or convergent, a zone of indiscernibility, wherein the borders vanish between the virtual and actual, expressed and expression, incorporeals and corporeals, sense in the proposition and event in states of affairs. Continuity conditions a fundamental indiscernibility but a heterogeneous one, a disjunctive synthesis that does not compromise or negotiate the heterogeneity of its terms. This divergent continuity is explored in Logic of Sense, through a dialogue with Leibniz’s syncategorematic account of calculus and Dedekind cut.

Dedekind Cuts, Moving Markers, and the Uncountability of ℝ

Summary We give short proofs that ℝ is uncountable directly from the definition of ℝ as the set of Dedekind cuts of ℚ .

THE TEMPORAL CONTINUUM

Abstract The continuum has been one of the most controversial topics in mathematics since the time of the Greeks. Some mathematicians, such as Euclid and Cantor, held the position that a line is composed of points, while others, like Aristotle, Weyl, and Brouwer, argued that a line is not composed of points but rather a matrix of a continued insertion of points. In spite of this disagreement on the structure of the continuum, they did distinguish the temporal line from the spatial line. In this paper, we argue that there is indeed a difference between the intuition of the spatial continuum and the intuition of the temporal continuum. The main primary aspect of the temporal continuum, in contrast with the spatial continuum, is the notion of orientation. The continuum has usually been mathematically modeled by Cauchy sequences and the Dedekind cuts. While in the first model, each point can be approximated by rational numbers, in the second one, that is not possible constructively. We argue that points on the temporal continuum cannot be approximated by rationals as a temporal point is a flow that sinks to the past. In our model, the continuum is a collection of constructive Dedekind cuts, and we define two topologies for temporal continuum: 1. oriented topology and 2. the ordinary topology. We prove that every total function from the oriented topological space to the ordinary one is continuous.

Navigating Number Systems for Computational Precision: An Analytical Exploration

The main objective of this article is to study all numbers that have occurred so far in the theoretical discussions were real (or complex) numbers in the strict mathematical sense. That is, they were to be conceived as infinite decimal fractions, or as Dedekind cuts. For the purposes of computation such numbers have to be approximated by real numbers of a rather special type, such as terminating decimal fractions, or other rational numbers. The present article is devoted to a study of the number systems that can be used for the purpose of computation.

Interplay between insertion of zeros and the complexity of Dedekind cuts

The topic of this paper is the study of the possibility of effective conversions between different representations of irrational numbers. We are interested in subrecursive computations, in which we disallow the use of unbounded search. It is well-known that the base expansion of an irrational number can be subrecursively computed from its Dedekind cut, but in the reverse direction this is not subrecursively possible. Thus it would be natural to try to understand how additional properties of the base expansion, such as the presence of specific patterns of zero digits, correlate with the complexity of the Dedekind cut. Our first result roughly states that if large parts of zero digits dominate over smaller parts of arbitrary digits and the base expansion has elementary complexity, then the Dedekind cut also has elementary complexity. Much more interesting is the case when one allows large parts of zero digits to alternate with large parts of arbitrary digits. We compare the complexity of the Dedekind cuts of an arbitrary irrational number and another irrational number, obtained by inserting long sequences of zero digits in its base expansion. We provide examples in which (1) both numbers have elementary Dedekind cuts, (2) the first number has complex Dedekind cut and elementary base expansion, but the second number has elementary Dedekind cut and (3) both numbers have complex Dedekind cuts and elementary base expansions. An open question remains, whether it is possible for the first number to have elementary Dedekind cut, while the second number has complex Dedekind cut.

Subrecursive incomparability of the graphs of standard and dual Baire sequences

Our main question of interest is the existence or the non-existence of a subrecursive reduction between different representations of the irrational numbers. For any representation, considered as a total function, we consider the characteristic function of its graph. The graph is computably equivalent to the function itself, but not subrecursively equivalent. In some cases, the graph of a representation is subrecursively equivalent to an already known representation, but in other cases it is a new representation. In the present paper we undertake a systematic study of the graphs of standard and dual Baire sequences. By combining our new results with the previously known results on the graph of the continued fraction, we obtain a total of eight new subrecursive degrees, which lie strictly between the Dedekind cut and the continued fraction.

Improvement of constructing real numbers by Dedekind cuts

In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the least-upper-bound property, as well as the density properties of rational cuts in the set of real numbers, is established. Later, when defining addition and multiplication and establishing relative properties, the definition of cut will not be used again. This makes the process simpler and easier to understand.

Computable irrational numbers with representations of surprising complexity

Cauchy sequences, Dedekind cuts, base-10 expansions and continued fractions are examples of well-known representations of irrational numbers. But there exist others, not so popular, which can be defined using various kinds of sum approximations and best approximations. In this paper we investigate the complexity of a number of such representations.For any fast-growing computable function f, we define an irrational number αf by using a series of reciprocals of powers of all primes. We prove that certain representations of αf are of low computational complexity (which does not depend on f), whereas others, apparently similar representations, can be of arbitrarily high computational complexity (which depends on f). The existence of computable numbers like αf allows us to prove new and non-trivial theorems on the computational complexity of representations without resorting to the standard computability-theoretic machinery involving enumerations and diagonalizations.In the paper we also show how to construct irrational numbers γ whose representations by a Cauchy sequences are of low computational complexity, but whose base-b expansion may be of arbitrarily high computational complexity for all bases b. Moreover, for any E2-irrational number α, there will be an E2-irrational number β, such that α+β has the complexity of γ. As a consequence, two numbers which have, let us say, base-10 expansions of low computational complexity, may add up to a number whose base-10 expansion is of arbitrarily high computational complexity. The same goes for representations by base-2 expansions, base-17 expansions, Dedekind cuts, continued fractions, and so on.

Space-time structure may be topological and not geometrical

In a previous effort we have created a framework that explains why topological structures naturally arise within a scientific theory; namely, they capture the requirements of experimental verification. This is particularly interesting because topological structures are at the foundation of geometrical structures, which play a fundamental role within modern mathematical physics. In this paper we will show a set of necessary and sufficient conditions under which those topological structures lead to real quantities and manifolds, which are a typical requirement for geometry. These conditions will provide a physically meaningful procedure that is the physical counter-part of the use of Dedekind cuts in mathematics. We then show that those conditions are unlikely to be met at Planck scale, leading to a breakdown of the concept of ordering. This would indicate that the mathematical structures required to describe space-time at that scale, while still topological, may not be geometrical.

On definable completeness for ordered fields

We show that there are 0-definably complete ordered fields which are not real closed. Therefore, the theory of definably with parameters complete ordered fields does not follow from the theory of 0-definably complete ordered fields. The mentioned completeness notions for ordered fields are the definable versions of completeness in the sense of Dedekind cuts. In earlier joint work, we had shown that it would become successively weakened if we just required nonexistence of definable regular gaps and then disallowing parameters. The result in this note shows reducing in the opposite order, at least one side is sharp.

Darboux calculus

We introduce a formalism to analyze partially defined functions between ordered sets. We show that our construction provides a uniform and conceptual approach to all the main definitions encountered in elementary real analysis including Dedekind cuts, limits and continuity.

On subrecursive representability of irrational numbers

We consider various ways to represent irrational numbers by subrecursive functions: via Cauchy sequences, Dedekind cuts, trace functions, several variants of sum approximations and continued fractions. Let S be a class of subrecursive functions. The set of irrational numbers that can be obtained with functions from S depends on the representation. We compare the sets obtained by the different representations.

On Closure Properties of Irrational and Transcendental Numbers under Addition and Multiplication

In the article 'There are Truth and Beauty in Undergraduate Mathematics Research’, the author posted a problem regarding the closure properties of irrational and transcendental numbers under addition and multiplication. In this study, we investigate the problem using elementary mathematical methods and provide a new approach to the closure properties of irrational numbers. Further, we also study the closure properties of transcendental numbers. KEYWORDS: Irrational numbers; Transcendental numbers; Dedekind cuts; Algebraic numbers

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that the castle is floating in midair. Halmos' realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians' concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson's framework is "unnecessary" but Henson and Keisler argue that Robinson's framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos' criticisms. Keywords: Archimedean axiom; bridge between discrete and continuous mathematics; hyperreals; incomparable quantities; indispensability; infinity; mathematical realism; Robinson.

Revisiting the construction of the real line

The most common constructions of the reals from the rationals attempt to “fill in the gaps,” using (equivalence classes of) Cauchy sequences of rationals or Dedekind cuts. Another technique, more familiar to logicians than to analysts, is to expand the set Q of rationals to the set QN of all sequences of rationals and then collapse back down in two stages. In the first stage, produce a structure X by identifying sequences that agree on a large set; in the second stage, further collapse X by removing its infinite elements, and collapsing its infinitesimals to 0. The latter approach, though less widely known, provides, in its intermediary construction X, an example of interesting pathologies. We show here that X is an example of a complete (even open-complete) ordered field that is not Archimedean. The field X showcases the significant role of the Archimedean property in the structure of R, as many of the fundamental properties of R are found to fail in X precisely because this property is absent, such as the Heine-Borel Theorem and the Bolzano-Weierstrass Theorem. Indeed, we show that the field X is not even metrizable. We conclude with a detailed proof, accessible to non-logicians, that the second stage of the construction succeeds in producing a complete Archimedean ordered field. Mathematics Subject Classification: 12J15, 26E35

Differential Calculus Based on the Double Contradiction

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction.

Ramsey for ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter on ω and each map is a Dedekind cut in the ultrapower . Blass has characterized, under , the cuts obtainable when is taken to be either a p‐point ultrafilter, a weakly‐Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorčević have introduced the topological Ramsey space . Associated to the space is a notion of Ramsey ultrafilter for generalizing the familiar notion of Ramsey ultrafilter on ω. We characterize, under , the cuts obtainable when is taken to be a Ramsey for ultrafilter and p is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with using almost‐reduction adjoins an ultrafilter which is Ramsey for . For such ultrafilters , Dobrinen and Todorčević have shown that the Rudin‐Keisler types of the p‐points within the Tukey type of consists of a strictly increasing chain of rapid p‐points of order type ω. We show that for any Rudin‐Keisler mapping between any two p‐points within the Tukey type of the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.

ON THE NUMBER OF DEDEKIND CUTS AND TWO-CARDINAL MODELS OF DEPENDENT THEORIES

For an infinite cardinal ${\it\kappa}$, let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$. It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all ${\it\kappa}$ and that $\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any ${\it\kappa}$ of uncountable cofinality. We prove however that $2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.

Why topology in the minimalist foundation must be pointfree

We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our two-level minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main reason is that in our foundation real numbers, either as Dedekind cuts or as Cauchy sequences, do not form a set.