scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

Relative randomness and real closed fields

2005 ◽  
Vol 70 (1) ◽  
pp. 319-330 ◽  
Author(s):  
Alexander Raichev
Keyword(s):  
Real Number ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Closed Fields ◽  
Master's Thesis ◽  
National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

scholarly journals Intersections of Real Closed Fields

1980 ◽  
Vol 32 (2) ◽  
pp. 431-440 ◽  
Author(s):  
Thomas C. Craven
Keyword(s):  
Galois Group ◽  
General Class ◽  
Entire Functions ◽  
Algebraic Closure ◽  
Careful Study ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Infinite Sequences

In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal , the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3].

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

Nonstandard Student Conceptions About Infinitesimals

2010 ◽  
Vol 41 (2) ◽  
pp. 117-146 ◽  
Author(s):  
Robert Ely
Keyword(s):  
Real Number ◽  
Nonstandard Analysis ◽  
Number Line ◽  
Real Numbers ◽  
Student Conceptions ◽  
The Real ◽  
New Perspective ◽  
Real Number Line ◽  
The Relationship

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

The Relationship Between Nontrivial Automorphisms and Order of Fields

1968 ◽  
Vol 61 (3) ◽  
pp. 246-250
Author(s):  
Victor Keiser
Keyword(s):  
Mathematics Teacher ◽  
Definite Article ◽  
Real Numbers ◽  
The Real ◽  
Nontrivial Automorphism ◽  
Ordered Fields ◽  
The Relationship

In the article “A Nontrivial Automorphism of the Field of Real Numbers” (THE MATHEMATICS TEACHER, December 1966), Robert F. Lawler defines operations * and Δ on a certain set F which he refers to as the field of real numbers. Before going further, let us point out that the use of the definite article in the phrase “the field of real numbers” is justified by the well-known theorem stating that any two complete ordered fields are isomorphic; it does not arise from the existence of some particular distinguished set of objects which we call the real numbers.

scholarly journals Models of true arithmetic are integer parts of models of real exponentation

2021 ◽  
Vol 13 ◽  
Author(s):  
Merlin Carl ◽  
Lothar Sebastian Krapp
Keyword(s):  
Multiplicative Group ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
Integer Part ◽  
Real Numbers ◽  
The Real ◽  
Positive Elements ◽  
Closed Fields ◽  
Schanuel's Conjecture

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

Frege on the Real Numbers

2019 ◽  
pp. 343-383
Author(s):  
Eric Snyder ◽  
Stewart Shapiro
Keyword(s):  
Light Intensity ◽  
Detailed Examination ◽  
Real Numbers ◽  
Detailed Review ◽  
The Real ◽  
Cardinal Numbers ◽  
Main Complaint ◽  
The Relationship

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.

Definable types in -minimal theories

1994 ◽  
Vol 59 (1) ◽  
pp. 185-198 ◽  
Author(s):  
David Marker ◽  
Charles I. Steinhorn
Keyword(s):  
Stability Theory ◽  
Conservative Extension ◽  
Absolute Value ◽  
First Order ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields ◽  
Models Of Arithmetic

Let L be a first order language. If M is an L-structure, let LM be the expansion of L obtained by adding constants for the elements of M.Definition. A type is definable if and only if for any L-formula , there is an LM-formula so that for all iff M ⊨ dθ(¯). The formula dθ is called the definition of θ.Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for M ≺ N, the property that every M-type realized in N is definable is equivalent to N being a conservative extension of M, whereDefinition. If M ≺ N, we say that N is a conservative extension of M if for any n and any LN -definable S ⊂ Nn, S ∩ Mn is LM-definable in M.Van den Dries [Dl] studied definable types over real closed fields and proved the following result.0.1 (van den Dries), (i) Every type over (R, +, -,0,1) is definable.(ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent:(a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F.(b) K is a conservative extension of F.

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