Dedekind cuts in C(X)

2011 ◽  
Vol 95 ◽  
pp. 287-297 ◽  
Author(s):  
Nicolae Dăneţ
Keyword(s):  
Dedekind Cuts

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

2015 ◽  
Vol 15 (4) ◽  
pp. 771-784 ◽  
Author(s):  
Artem Chernikov ◽  
Saharon Shelah
Keyword(s):  
Infinite Cardinal ◽  
Linear Orders ◽  
First Order ◽  
Dedekind Cuts

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.

Model completeness of o-minimal structures expanded by Dedekind cuts

2005 ◽  
Vol 70 (1) ◽  
pp. 29-60 ◽  
Author(s):  
Marcus Tressl
Keyword(s):  
Valuation Ring ◽  
Ordered Set ◽  
Real Closed Field ◽  
Complete Theory ◽  
Closed Field ◽  
Model Completeness ◽  
Unary Predicate ◽  
Minimal Structure ◽  
Totally Ordered Set ◽  
Dedekind Cuts

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.

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

2016 ◽  
Vol 13 (3) ◽  
Author(s):  
Shekh Zahid ◽  
Prasanta Ray
Keyword(s):  
Algebraic Numbers ◽  
Undergraduate Mathematics ◽  
Mathematical Methods ◽  
Closure Properties ◽  
New Approach ◽  
Irrational Numbers ◽  
Transcendental Numbers ◽  
Dedekind Cuts ◽  
Mathematics Research

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

scholarly journals On constructing completions

2005 ◽  
Vol 70 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Laura Crosilla ◽  
Hajime Ishihara ◽  
Peter Schuster
Keyword(s):  
Set Theory ◽  
Metric Space ◽  
Ordered Set ◽  
Proper Class ◽  
Original Space ◽  
Full Form ◽  
Dedekind Cuts ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

AbstractThe Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

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