Continuity in Intuitionism

2020 ◽  
pp. 299-327
Author(s):  
Charles McCarty
Keyword(s):  
Real Number ◽  
Number System ◽  
Real Numbers ◽  
Continuity Theorem ◽  
The Real ◽  
Real Number System

The chapter features, first, a critical presentation of Brouwer’s intuitionistic doctrines concerning logic, the real numbers, and continuity in the real number system, including his Principle for Numbers and Continuity Theorem. This is followed by a parallel examination of Hermann Weyl’s quasi-intuitionistic views on logic, continuity, and the real number system, views inspired by (but grossly misrepresenting) ideas of Brouwer. The whole business wraps up with an attempt to place Brouwer’s and Weyl’s efforts within the trajectory of informed thinking, during the late 19th and early 20th centuries, on the subjects of continua, magnitudes, and quantities.

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 $\mathbb N$.}

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.

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 A New Set Theory for Analysis

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 $\mathbb N$.}

scholarly journals SEL Series Expansion and Generalized Model Construction for the Real Number System via Series of Rationals

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Patiwat Singthongla ◽  
Narakorn Rompurk Kanasri
Keyword(s):  
Real Number ◽  
Series Expansion ◽  
Infinite Series ◽  
Number System ◽  
Model Construction ◽  
Series Expansions ◽  
Real Numbers ◽  
Generalized Model ◽  
The Real ◽  
Real Number System

We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and Lüroth series expansions. Using series of rationals, a generalized model for the real number system is also constructed.

scholarly journals The cognitive process of extended trans students in understanding the real number system

2020 ◽  
Vol 1470 ◽  
pp. 012070
Author(s):  
Dewi Herawaty ◽  
Wahyu Widada ◽  
Iran Sairan ◽  
Fizi Herdian ◽  
Khathibul U Z Nugroho ◽  
...  
Keyword(s):  
Real Number ◽  
Cognitive Process ◽  
Number System ◽  
The Real ◽  
Real Number System
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