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Advances in Pure Mathematics, 2016, 6, 593-607 Published Online August 2016 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2016.69048

How to cite this paper: Escultura, E.E. (2016) The Constructivist Real Number System. Advances in Pure Mathematics, 6, 593-607. http://dx.doi.org/10.4236/apm.2016.69048

The Constructivist Real Number System Edgar E. Escultura GVP-Professor V. Lakshmikantham Institute for Advanced Studies, GVP College of Engineering, JNT University Kakinada Madhurawada, Vishakhapatnam, India

Received 16 May 2016; accepted 14 August 2016; published 17 August 2016

Copyright 2016 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract The paper summarizes the contributions of the three philosophies of mathematicslogicism, in-tuitionism-constructivism (constructivism for short) and formalism and their rectificationwhich constitute the new foundations of mathematics. The critique of the traditional foundations of ma-thematics reveals a number of errors including inconsistency (contradiction or paradox) and un-defined and vacuous concepts which fall under ambiguity. Critique of the real and complex num-ber systems reveals similar defects all of which are responsible not only for the unsolved long standing problems of foundations but also of traditional mathematics such as the 379-year-old Fermats last theorem (FLT) and 274-year-old Goldbachs conjecture. These two problems require rectification of these defects before they can be resolved. One of the major defects is the inconsis-tency of the field axioms of the real number system with the construction of a counterexample to the trichotomy axiom that proved it and the real number system false and at the same time not li-nearly ordered. Indeed, the rectification yields the new foundations of mathematics, constructivist real number system and complex vector plane the last mathematical space being the rectification of the complex real number system. FLT is resolved by a counterexample that proves it false and the Goldbachs conjecture has been proved both in the constructivist real number system and the new real number system. The latter gives to two mathematical structures or toolsgeneralized integral and generalized physical fractal. The rectification of foundations yields the resolution of problem 1 and the solution of problem 6 of Hilberts 23 problems.

Keywords Axiom of Choice, Banach-Tarski Paradox, Continuum, Dark Number, Decimal Integer, D-Sequence, G-Norm, G-Sequence, Nonterminating Decimal, Russell Antimony, Self-Reference, Trichotomy Axiom

1. Introduction The paper summarizes the contributions of the three schools of thought or philosophies of mathematicslogi-

http://www.scirp.org/journal/apmhttp://dx.doi.org/10.4236/apm.2016.69048http://dx.doi.org/10.4236/apm.2016.69048http://www.scirp.orghttp://creativecommons.org/licenses/by/4.0/

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cism [1], intuitionism-constructivism (constructivism for short) [2] and formalism [3] and their rectification that constitute the new foundations of mathematics [4]. Their critique-rectification undertaken in [4] including the critique-rectification of the real number system [5] summarized below is the main contribution of [4]. The main contribution of logicism is the discovery of the Russell Antimony [6] and that of intuitionism the rejection of the indirect proof as a case of self-reference [4] [7]. L. E. J. Brouwers requirement not only that every con-cept and conclusion of a theorem are not vacuous (i.e., they exist) but also constructible or demonstrable ad-vanced intuitionism to the philosophy of constructivism [8] [9]. Mathematics made another major advance when David Hilbert discovered a century ago that the concepts of individual thought being inaccessible to others are ambiguous and required that mathematical concepts be objects in the real world called concepts such as words, symbols, numerals, figures and chess pieces subject to consistent basic premises or axioms which gave rise to the philosophy of formalism [3]. This resolves the problem with the equation 1 = 0.99, which is wrong and akin to the statement apple = orange, since 1 and 0.99 are distinct objects. The problem is resolved by defining = as the relation the same as. Then 1 0.99 and = is an equivalence relation and therefore satisfies the identity, reflexivity and transitivity properties which need not be taken as axioms.

Until today, mathematicians have not grasped the significance of Hilberts discovery to the extent that all textbooks in mathematics assume 1 = 0.99 When this issue was raised in internet forums in 1997, especially, SciMath, it sparked a howl of protest and controversy with a whisk of name calling that lasted over a decade and spilled over into many websites.

2. Critique-Rectification of Traditional Mathematics The full critique-rectification of traditional mathematics and its foundations done in [4] completes the construc-tion of the new mathematics and its foundations started by Hilbert. While he required the concepts of a mathe-matical system or space to be subject to consistent axioms or basic premises to avoid contradiction or paradox he still left a big hole in mathematics with admission of ambiguity, e.g., undefined and vacuous concepts, and re-tention of the indirect proof which has been rejected by constructivism for compelling reason. We complete the rectification by rejecting the indirect proof once again and requiring that every concept be defined by the axioms and their properties and behavior be derived from or supported by them.

One of the field axioms of the real number system [10], the trichotomy axiom, is false according to the fol-lowing counterexample [5]:

Given two rationals x, y we can tell if x < y or x > y. Even then, we cannot line up all the rationals on the real line under the ordering < due to the ambiguity of the infinite number of rationals between any two given ration-als and this is due to the ambiguity of the concept infinity. However, we can proceed with the following scenario: start with a certain rational interval [A, B] with A < C < B, and find a nested sequence of rational intervals [An, Bn], with n nA A C B B< < < < , for each 1, 2,3, ,n n= . At each stage, we want to make sure that

10 nnA C< , 10 nnB C

> + , and [ ] [ ] [ ]1 1, , ,n n n nA B A B A B . Since the nonterminating decimal C is defined to be the limit of a sequence of rationals (from the left and

from the right), we can choose the end points An, Bn, of intervals [An, Bn] as members of two sequences {An}, {Bn} where {An} is a monotonic increasing sequence and {Bn} a monotonic decreasing sequence of rationalssa-tisfying 1 2 2 1n nA A A A C B B B B < < , and for each n, 10

nnC A

and 10 nnB C .

This process can be continued as long as we are able to identify An, Bn to be such that An < C < Bn, i.e. as far as we know the decimal representation of C with its n decimal digits. It cannot be taken further since we are unable to find An+1, Bn+1 with error of 10(n+1) and establish An+1 < C < Bn+1, with C being known only to n places. No matter how large the number n is, we still have the disadvantage of not getting the next interval [An+1, Bn+1]. Consequently, we have to acknowledge the inherent trouble involved with understanding and dealing with non-terminating decimals and with the concept of infinity. This example shows that the real number system has no ordering under the relation < and the trichotomy axiom which says, given two real numbers x, y, only one of the following holds: x < y, x = y, x > y, is unverifiable.

The counterexample says that the real number system is not linearly ordered by the relation

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construction of a mathematical space the choice of the axioms is not complete until every concept is defined. While the choice of axioms is arbitrary depending on what the mathematical space is intended for, once chosen the mathematical space becomes deductive, i.e., every theorem follows from the axioms.

2) Avoidance of vacuous concept such as, root of the equation x2 + 1 = 0, denoted by 1i = which yields the contradiction, 1 1 1 1 ii i = = = = , from which follows 1 = 0 and I = 0. The concept i is vacuous be-cause the above equation has no root. Ill-defined concepts are vacuous.

3) Avoidance of infinity that cannot be contained by replacing its traditional definition with the concept inex-haustibility as its essential property so that an infinite set cannot be contained in a finite set and if one tries to put its elements in a finite set some element will be left out at every step. This property of infinite set invalidates the axiom of choice when applied to infinite set [10] [11]. It makes the concept irrational as nonperiodic nontermi-nating decimal ambiguous, ill-defined since this property is not verifiable. The same argument collapses Can-tors diagonal method [12]. Set theory [13] and the proof of the Banach-Tarski paradox are flawed [14].

4) Avoidance of self-reference [4], the source of the Russell paradoxes [15] [16] of which the most famous is the Russell antimony [6]. Russells remedy for his paradoxes is his type theory [17] that rejects them. The avoidance of self-reference collapses nondenumerable and non-measurable sets and the universal and power sets. They do not exist. The indirect proof is self-referent and is also rejected.

Requirement 1) invalidates proof of theorem involving concepts from two distinct mathematical spaces, e.g., Gdels incompleteness theorems [18]. By rejecting the indi