On virtual classes and real numbers

1950 ◽  
Vol 15 (2) ◽  
pp. 131-134
Author(s):  
R. M. Martin
Keyword(s):  
Standard Procedure ◽  
Homogeneous System ◽  
Alternative Methods ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Virtual Class ◽  
Virtual Classes ◽  
Main Ideas

In a simple, applied functional calculus of first order (i.e., one admitting no functional variables but at least one functional constant), abstracts or schematic expressions may be introduced to play the role of variables over designatable sets or classes. The entities or quasi-entities designated or quasi-designated by such abstracts may be called, following Quine, virtual classes and relations. The notion of virtual class is always relative to a given formalism and depends upon what functional constants are taken as primitive. The first explicit introduction of a general notation for virtual classes (relative to a given formalism) appears to be D4.1 of the author's A homogeneous system for formal logic. That paper develops a system admitting only individuals as values for variables and is adequate for the theory of general recursive functions of natural numbers. Numbers and functions are in fact identified with certain kinds of virtual classes and relations.In the present paper it will be shown how certain portions of the theory of real numbers can be constructed upon the basis of the theory of virtual classes and relations of H.L.The method of building up the real numbers to be employed is essentially an adaptation of standard procedure. Although the main ideas underlying this method are well known, the mirroring of these ideas within the framework of the restricted concepts admitted here presents possibly some novelty. In particular, a basis for the real numbers is provided which in no way admits classes or relations or other "abstract" objects as values for variables. Presupposing the natural numbers, the essential steps are to construct the simple rationals as virtual dyadic relations between natural numbers, to construct the generalized or signed rationals as virtual tetradic relations among natural numbers, and then to formulate a notation for real numbers as virtual classes (of a certain kind) of generalized rationals. Of course, there are several alternative methods. This procedure, however, appears to correspond more to the usual one.

scholarly journals CREATING SENSATION OF LEARNING IN CLASSROOM: USING 'GATHER TOWN' PLATFORM VIDEO GAME-STYLE FOR VIRTUAL CLASSROOM

2021 ◽  
Vol 6 (2) ◽  
pp. 30-43
Author(s):  
Tira Nur Fitria
Keyword(s):  
Language Learning ◽  
Video Game ◽  
English Language ◽  
Virtual Classroom ◽  
Work Activities ◽  
The Real ◽  
Virtual Meeting ◽  
Virtual Class ◽  
Virtual Classes ◽  
Result Analysis

Abstract: Virtual classes have now become commonplace during the COVID-19 pandemic. As time goes by, applications for virtual meetings continue to appear, one of which is Gather Town, which is now widely used and is a tight competitor to Zoom Meeting and Google Meet. Gather Town is a virtual meeting platform designed like a video game. This research is to implement the use of the 'Gather Town' game platform and to find out the student’s perception during the implementation and simulation of Gather Town application as an alternative platform in creating a sensation of English Language Learning (ELL) in the real classroom through virtual class during the pandemic COVID-19. This study uses descriptive qualitative research. The result analysis from observation and interview show that Gather Town has graphics similar to the Harvest Moon game, where students can play one character and can write their name on the top so that the lecturer can see which students are present. The room is designed similar to a classroom, where the lecturer's desk is at the front of the classroom. The virtual classroom also has chairs that are neatly lined up like classrooms in the real world. Then when doing group assignments, the student characters will gather at the same table as in a real classroom. They also carry out group work activities as if they were in the classroom. Each group sat in a circle and discussed with one another. The current game may be an alternative design in a virtual classroom. Abstrak: Kelas virtual kini menjadi hal biasa selama pandemi COVID-19. Seiring berjalannya waktu, aplikasi untuk virtual meeting terus bermunculan, salah satunya adalah Gather Town yang kini banyak digunakan dan menjadi pesaing ketat Zoom Meeting dan Google Meet. Gather Town adalah platform pertemuan virtual yang dirancang seperti gim video. Penelitian ini bertujuan untuk mensimulasikan penggunaan platform game 'Gather Town' dan mengetahui persepsi siswa selama simulasi aplikasi Gather Town sebagai platform alternatif dalam menciptakan sensasi Pembelajaran Bahasa Inggris di kelas nyata melalui virtual. kelas selama pandemi COVID-19. Penelitian ini merupakan penelitian kualitatif deskriptif. Hasil analisis dari observasi dan wawancara menunjukkan bahwa Gather Town memiliki grafik yang mirip dengan game Harvest Moon, dimana mahasiswa dapat memainkan satu karakter dan dapat menuliskan namanya di bagian atas sehingga dosen dapat melihat mahasiswa mana yang hadir. Tidak hanya itu, ruangannya didesain mirip dengan ruang kelas, dimana meja dosen berada di bagian depan kelas. Ruang kelas virtual juga memiliki kursi yang berjejer rapi seperti ruang kelas di dunia nyata. Kemudian saat mengerjakan tugas kelompok, karakter siswa akan berkumpul di meja yang sama seperti di ruang kelas yang sebenarnya. Mereka juga melakukan kegiatan kerja kelompok seolah-olah berada di dalam kelas. Setiap kelompok duduk melingkar dan berdiskusi satu sama lain. Game saat ini dapat menjadi alternatif desain di kelas virtual.

Proper Placement of Derived Classes in the Class Hierarchy

2005 ◽  
pp. 486-492
Author(s):  
Reda Alhajj ◽  
Faruk Polat
Keyword(s):  
Object Oriented ◽  
Homogeneous System ◽  
Class Hierarchy ◽  
Virtual Class ◽  
Virtual Classes ◽  
Class Definition ◽  
Current Database

Users may derive new classes by defining views1 based on the current database contents. Some virtual classes are classified as brothers of existing classes, and others are either superclasses or subclasses of existing base and virtual classes. A base class is defined directly by the user using class definition constructs. A virtual class is classified as a brother of another class if it is derived from the latter class via a selection. To have a homogeneous system, virtual classes must be treated as first-class citizens in an object-oriented model.

scholarly journals The Diagonalization Paradox

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Open Interval ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Diagonal Argument ◽  
One To One

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).

scholarly journals Russian Dolls

1978 ◽  
Vol 21 (2) ◽  
pp. 237-240 ◽  
Author(s):  
J. B. Wilker
Keyword(s):  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Strengthened Version

In an earlier number of this Bulletin, P. Erdös [1] posed the following problem. “For each line ℓ of the plane, Aℓ is a segment of ℓ. Show that the set ∪ℓAℓ contains the sides of a triangle.” One objective of this paper is to prove a strengthened version of this result in iV-dimensions. As usual denotes the cardinality of the natural numbers and c, the cardinality of the real numbers.

scholarly journals A New Set Theory for Analysis

2019 ◽  
Author(s):  
Juan Ramirez
Keyword(s):  
Real Number ◽  
Number System ◽  
Number Line ◽  
Physical Models ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Real Number System

We present the real number system as a natural generalization of the natural numbers. First, we prove the co-finite topology, $Cof(\mathbb N)$, is isomorphic to the natural numbers. Then, we generalize these results to describe the continuum $[0,1]$. Then we prove the power set $2^{\mathbb Z}$ contains a subset isomorphic to the non-negative real numbers, with all its defining structure of operations and order. Finally, we provide two different constructions of the entire real number line. We see that the power set $2^{\mathbb N}$ can be given the defining structure of $\mathbb R$. The constructions here provided give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. The supremum and infimum are explicitly constructed by means of a well defined algorithm that ends in denumerable steps. In section 5 we give evidence our construction of $\mathbb N$ and $\mathbb R$ are canonical; these constructions are as natural as possible. In the same section, we propose a new axiomatic basis for analysis. In the last section we provide a series of graphic representations and physical models that can be used to represent the real number system. We conclude that the system of real numbers is completely defined by the order structure of $\mathbb N$.}

scholarly journals The Diagonalization Paradox - Expanded

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
One To One ◽  
The One

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).

Mathematics Developed

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Number System ◽  
Scientific Practices ◽  
Natural Numbers ◽  
Real Numbers ◽  
Sir Isaac Newton ◽  
The Real ◽  
Isaac Newton ◽  
Number Systems ◽  
Real Number System

This chapter considers two crucial shifts in mathematical knowledge: the natural numbers ℕ and the real number system ℝ. ℝ has proved to serve together with the natural numbers ℕ as one of the two core structures of mathematics; together they are what Solomon Feferman described as “the sine qua non of our subject, both pure and applied.” Indeed, nobody can claim to have a basic grasp of mathematics without mastery of the central elements in the theory of both number systems. The chapter examines related theories and conceptions about real numbers, with particular emphasis on the work of J. H. Lambert and Sir Isaac Newton. It also discusses various conceptions of the number continuum, assumptions about simple infinity and arbitrary infinity, and the development of mathematics in relation to the real numbers. Finally, it reflects on the link between mathematical hypotheses and scientific practices.

What's Between Q and R

1967 ◽  
Vol 60 (4) ◽  
pp. 308-314
Author(s):  
James Fey
Keyword(s):  
Mathematics Instruction ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Valuable Insight ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Number Systems ◽  
Insight Into ◽  
Structure Properties

Among the objectives of school mathematics instruction, one of the most important is to develop understanding of the structure, properties, and evolution of the number systems. The student who knows the need for, and the technique of, each extension from the natural numbers through the complex numbers has a valuable insight into mathematics. Of the steps in the development, that from the rational numbers to the real numbers is the trickiest.

scholarly journals A New Set Theory for Analysis

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 31
Author(s):  
Juan Ramírez
Keyword(s):  
Real Number ◽  
Number System ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Universe Of Sets ◽  
The Universe ◽  
Real Number System

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

scholarly journals FREGE’S CONSTRAINT AND THE NATURE OF FREGE’S FOUNDATIONAL PROGRAM

2018 ◽  
Vol 12 (1) ◽  
pp. 97-143 ◽  
Author(s):  
MARCO PANZA ◽  
ANDREA SERENI
Keyword(s):  
Mathematical Theory ◽  
Essential Elements ◽  
Real Analysis ◽  
Natural Numbers ◽  
Real Numbers ◽  
Crispin Wright ◽  
Definition Of ◽  
Image Position ◽  
Shed Light

AbstractRecent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5).

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