A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

scholarly journals Some elementary results in intuitionistic model theory

1996 ◽  
Vol 61 (3) ◽  
pp. 745-767
Author(s):  
Wim Veldman ◽  
Frank Waaldijk
Keyword(s):  
Model Theory ◽  
Additive Group ◽  
Real Numbers ◽  
Elementary Model ◽  
The Real

AbstractWe establish constructive refinements of several well-known theorems in elementary model theory. The additive group of the real numbers may be embedded elementarily into the additive group of pairs of real numbers, constructively as well as classically.

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.

Wreath Products of Nonoverlapping Lattice Ordered Groups

1975 ◽  
Vol 17 (5) ◽  
pp. 713-722 ◽  
Author(s):  
John A. Read
Keyword(s):  
Wreath Product ◽  
Wreath Products ◽  
General Question ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Totally Ordered Group ◽  
Product Of Subgroups

One of the fundamental tools in the theory of totally ordered groups is Hahn’s Theorem (a detailed discussion may be found in Fuchs [3]), which asserts, roughly, that every abelian totally ordered group can be embedded in a lexicographically ordered (unrestricted) direct sum of copies of the ordered group of real numbers. Almost any general question regarding the structure of abelian totally ordered groups can be answered by reference to Hahn’s theorem. For the class of nonabelian totally ordered groups, a theorem which parallels Hahn’s Theorem is given in [5], and states that each totally ordered group can be o-embedded in an ordered wreath product of subgroups of the real numbers. In order to extend this theorem to include an “if and only if” statement, one must consider lattice ordered groups, as an ordered wreath product of subgroups of the real numbers is, in general, not totally-ordered, but lattice ordered.

Additive Functionals on Groups

1962 ◽  
Vol 58 (2) ◽  
pp. 196-205 ◽  
Author(s):  
Allan Hayes
Keyword(s):  
Linear Space ◽  
Linear Functional ◽  
Linear Subspace ◽  
Additive Group ◽  
Real Numbers ◽  
The Real ◽  
Additive Functionals

The kernel of a non-trivial linear functional φ on a linear space E is a maximal proper linear subspace of E which determines φ up to a non-zero multiple. Does a similar result hold for homomorphisms of a group G into the additive group R of the real numbers?

THE 2017 AMENDMENTS TO THE LAW REFORM (MARRIAGE AND DIVORCE) ACT 1976: A MILESTONE OR A STONE’S THROW IN THE DEVELOPMENT OF MALAYSIAN FAMILY LAW?

2020 ◽  
Vol 28 (2) ◽  
pp. 449-479
Author(s):  
Sridevi Thambapillay
Keyword(s):  
Child Custody ◽  
Family Law ◽  
Law Reform ◽  
The Real ◽  
The Past ◽  
Marriage And Divorce ◽  
Conversion To Islam ◽  
Family Issues ◽  
The Law ◽  
Muslim Family

The Law Reform (Marriage and Divorce) Act 1976 (LRA) which was passed in 1976 and came into force on 1st March 1982, standardized the laws concerning non-Muslim family matters. Many family issues concerning non-Muslim have emerged ever since, the most important being the effects of unilateral conversion to Islam by one of the parties to the marriage. There has been a lot of public hue and cry for amendments to be made to the LRA. After much deliberation, the Malaysian Parliament finally passed the amendments to the LRA in October 2017, which came into force in December 2018. Although the amendments have addressed selected family law issues, the most important amendment on child custody in a unilateral conversion to Islam was dropped from the Bill at the last minute. Howsoever, at the end of the day, the real question that needs to be addressed is whether the amendments have resolved the major issues that have arisen over the past four decades? Hence, the purpose of this article is as follows: first, to examine the brief background to the passing of the LRA, secondly, to analyse the 2017 amendments, thirdly, to identify the weaknesses that still exist in the LRA, and finally, to suggest recommendations to overcome these weaknesses by comparing the Malaysian position with the Singaporean position. In conclusion, it is submitted that despite the recent amendments to the LRA, much needs to be done to overcome all the remaining issues that have still not been addressed.

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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.

The Law Enforcement Weakness of Songs and Musics Copyright Crimes

2021 ◽  
Vol 3 (1) ◽  
pp. 139-154
Author(s):  
Edi Tuahta Putra Saragih ◽  
Muhammad Citra Ramadhan ◽  
Isnaini Isnaini
Keyword(s):  
Law Enforcement ◽  
Research Method ◽  
Copyright Infringement ◽  
Copyright Law ◽  
Cultural Factor ◽  
Legal Research ◽  
The Real ◽  
The Law ◽  
Real Picture

This research aimed to: (a) obtain the forms of copyright infringement of songs and/or music (with or without lyrics); (b) understand the role of the police, in this case the Police Precinct, in the law enforcement; (c) identify the factors that influenced the law enforcement. The research method used the normative-empirical legal research, with the initial stages of specifying norms in order to get the proper picture, and then specifying empirical events in order to get the real picture. The research results showed several matters: 1) The forms of copyright infringement of songs and/or music (with or without lyrics) found included: the distribution of the works or the copies, the performances of the works, and the announcements of the works; 2) Police Precinct did notultimately carry out their role as a law enforcer for the copyright infringement of songs and/or music (with or without lyrics); and 3) The factors that influenced the law enforcement on the copyright infringement of songs and/or music (with or without lyrics), namely: legislation factor, in the matter of complaint offenses; law enforcement factor, in terms of the capacity of members; less supportive factor of facilities and infrastructure; legal awareness factor, in the problem of the lack of legal counseling; and cultural factor, related to the differences in norms in the copyright law between those in society and those in regulations. 

On expressible sets and p-adic numbers

2011 ◽  
Vol 54 (2) ◽  
pp. 411-422
Author(s):  
Jaroslav Hančl ◽  
Radhakrishnan Nair ◽  
Simona Pulcerova ◽  
Jan Šustek
Keyword(s):  
Haar Measure ◽  
Real Numbers ◽  
The Real ◽  
Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

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