winter school in abstract analysis 2013 section set theory & topology

11Winter school in Abstract Winter school in Abstract Analysis 2013 Analysis 2013

Plekhanov Russia University of Plekhanov Russia University of EconomicsEconomics

Winter School in Abstract Analysis 2013Winter School in Abstract Analysis 2013section Set Theory & Topologysection Set Theory & Topology

Plekhanov Russian University of Economics

The innovative structure of point sets in terms of the next point principle

26th Jan — 2nd Feb 2013

Sidorov O.V.



MotivationMotivationMain notions of topology were extended from these of Main notions of topology were extended from these of arithmetic and geometry arithmetic and geometry

Geometrical notions and methods are clear, intuitive Geometrical notions and methods are clear, intuitive and natural as they are associated with reality.and natural as they are associated with reality.

Structure of Structure of point sets in particular sets on real line in particular sets on real line can be better analyzed in terms of Geometry.can be better analyzed in terms of Geometry.

Constructing real numbers in terms of arithmetic Constructing real numbers in terms of arithmetic resulted in unsolvable problems.resulted in unsolvable problems.

For description of real space-time structure in present For description of real space-time structure in present axiomatically generated real numbers are not sufficient axiomatically generated real numbers are not sufficient



HistoryHistory

For a long time Geometry while it used For a long time Geometry while it used Euclid’ methods was more advanced science Euclid’ methods was more advanced science than Arithmetic. And even than Arithmetic. And even Plato placed above Plato placed above the door of his the door of his AcademiaAcademia the words, "Let no the words, "Let no one ignorant of geometry enter here“one ignorant of geometry enter here“

Natural, Rational, and Real numbers appeared Natural, Rational, and Real numbers appeared due to needs of Geometrydue to needs of Geometry



HistoryHistoryHowever in the end of 19However in the end of 19thth century the century the correctnesscorrectness of of foundations of Geometry wasfoundations of Geometry was subjected to subjected to critiquecritiqueVVerification erification of the of the correctnesscorrectness was made by methods was made by methods of Arythmeticof Arythmetic«The proof of the compatibility of the axioms in geometry is effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers»Hilbert D. Mathematical Problems



HistoryHistory

In consequence of implementation of Hilbert’ program Geometry and its methods were assigned Geometry and its methods were assigned auxilary role.auxilary role.So G. Cantor was in favor of the actual infinity and So G. Cantor was in favor of the actual infinity and against the actual infinitesimal considered as a against the actual infinitesimal considered as a segment of the rectilinear continuum because claimed segment of the rectilinear continuum because claimed to have proved the impossibility of the actual to have proved the impossibility of the actual infinitesimal as a segment of the rectilinear infinitesimal as a segment of the rectilinear continuum, using his transfinite numbers.continuum, using his transfinite numbers.G. Cantor Mitteilungen zur Lehre von transfiniten, G. Cantor Mitteilungen zur Lehre von transfiniten, Zeitschrift fur Philosophie und philosophische Kritik, Zeitschrift fur Philosophie und philosophische Kritik, 91, pp 81-225, 92, pp.242-26591, pp 81-225, 92, pp.242-265



Real lineReal line

«We assume that you are familiar with the «We assume that you are familiar with the geometric interpretation of the real numbers as geometric interpretation of the real numbers as points on a line. …. Henceforth, we will use the points on a line. …. Henceforth, we will use the terms real number system and real line terms real number system and real line synonymously and denote both by the symbol R; synonymously and denote both by the symbol R; also, we will often refer to a real number as a also, we will often refer to a real number as a point (on the real line) ». p.19point (on the real line) ». p.19

Trench W.F. Introduction to Real Analysis Trench W.F. Introduction to Real Analysis (Prentice Hall), 2003(Prentice Hall), 2003



Real lineReal line

«The reader is undoubtedly aware that the set of real «The reader is undoubtedly aware that the set of real numbers R may be represented as a horizontal line numbers R may be represented as a horizontal line called the real line. Each point on the real line called the real line. Each point on the real line corresponds to a unique real number and each real corresponds to a unique real number and each real number corresponds to a unique point on the line. We number corresponds to a unique point on the line. We shall speak interchangeably of the set of real numbers shall speak interchangeably of the set of real numbers R and the real line R» p.2R and the real line R» p.2

Stancl D. L. Real analysis with point-set topology. Stancl D. L. Real analysis with point-set topology. MARCEL DEKKER, INC. 1987MARCEL DEKKER, INC. 1987



Real lineReal line

«A real line«A real line is a linearly ordered field is a linearly ordered field R R satisfying the Bolzano Principle: every non-satisfying the Bolzano Principle: every non-empty subset of R bounded from above has a empty subset of R bounded from above has a supremum» p.39supremum» p.39

Bukovský L. The Structure of the Real Line, Bukovský L. The Structure of the Real Line, Springer Basel AG, 2011Springer Basel AG, 2011



Structure of Real lineStructure of Real line

3

Dots are rationals, gaps are irrationals 3 as a gap in the rationals



Intervals in R

A closed interval in R

An open interval in R



SummarySummaryThere is There is one-to-one correspondence one-to-one correspondence between between points on the real line and real numbers in Rpoints on the real line and real numbers in R

The real line is a homogeneous and isotropic linear The real line is a homogeneous and isotropic linear ordered infinite set of pointsordered infinite set of points

Structure of real line and intervals on it Structure of real line and intervals on it presupposepresupposes possibility of existence of next points s possibility of existence of next points for each given pointfor each given point

Properties of completeness and denseness of real Properties of completeness and denseness of real numbers are incompatible with existence of next numbers are incompatible with existence of next points for each given pointpoints for each given point



Main theoryMain theory The Principle of next point

«Any point of real line interval containing not less than 2 points has at least one next point»

Example of real line interval is shown in the figure

Proof. Consider the real line. It is a homogeneous and isotropicIt is a homogeneous and isotropic linear ordered infinite set of points, i.e actual infinity of points.linear ordered infinite set of points, i.e actual infinity of points.Property of Property of homogeneityhomogeneity implies that all the points are equivalent. Then choose any given point.(i) If there is no one next point from both sides of the given point, then the point is isolated. As all point of the real line are equivalent, then all points are also isolated. But this contradicts the supposition, that there are no gaps on the real line. Thus any point of real line has two next points, one from each side.



Main theoryMain theory

(ii) If there is no next point from one side of the given point, then there is no next point from another side of the given point, andagain the point is isolated. Further proofs is as (i). In the case of interval One chooses two distinct points and Consideration of the case is not of difficulty.

Corollary 1 Open interval has end points and topologically is equivalent to close interval.

Corollary 2 Real numbers a infinitely but not unlimitedly divisible.



ConclusionConclusion

Geometrical and Arithmetic interpretation Geometrical and Arithmetic interpretation of real numbers are not consistentof real numbers are not consistent

Another different model of real numbers Another different model of real numbers can be constructed in terms of Geometry can be constructed in terms of Geometry



Thank YouThank You

winter school in abstract analysis 2013 section set theory & topology

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