Russian Dolls

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.

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The Diagonalization Paradox

10.31219/osf.io/zdt9v ◽

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).

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A New Set Theory for Analysis

10.20944/preprints201901.0089.v2 ◽

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 $\mathbb N$.}

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The Diagonalization Paradox - Expanded

10.31219/osf.io/9hwrg ◽

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).

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Mathematics Developed

Mathematical Knowledge and the Interplay of Practices ◽

10.23943/princeton/9780691167510.003.0008 ◽

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.

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What's Between Q and R

Mathematics Teacher ◽

10.5951/mt.60.4.0308 ◽

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.

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A New Set Theory for Analysis

Axioms ◽

10.3390/axioms8010031 ◽

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.

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An Axiomatic Theory of Ordinal Numbers

Nagoya Mathematical Journal ◽

10.1017/s0027763000002324 ◽

1961 ◽

Vol 18 ◽

pp. 193-228

Author(s):

Toshio Umezawa

Keyword(s):

Binary Relation ◽

Formal Theory ◽

Proper Class ◽

Natural Numbers ◽

Axiomatic Theory ◽

Real Numbers ◽

The Real ◽

Ordinal Numbers ◽

The Difference

In this paper, a formal theory of ordinal numbers is developed on an axiomatic basis whose details are described in § 1. Our primitive notions are set, class, collection, a binary relation ∈, and collection formation {!}. Sets and classes in our theory play similar roles as sets and classes respectively in Gödel [1] except the difference that an element of a class is a class but not necessarily a set. A new notion, introduced into our theory is that of collections. A collection relates to a class, just as a class relates to a set in von Neumann’s theory. That is; a set is a class and a class is a collection but the converses are not generally the case. For example, all the natural numbers, all the real numbers etc. constitute sets, the ordinal numbers which are sets constitute a proper class, and the totality of ordinal numbers as well as that of all classes are proper collections. These relations are described by axiom group (A).

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A New Set Theory for Analysis

10.20944/preprints201901.0089.v1 ◽

2019 ◽

Author(s):

Juan P. 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 $\mathbb N$.}

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Shuffling of Linear Orders

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-032-1 ◽

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.

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