Winter School in Abstract Analysis 2013 section Set Theory & Topology

1. Winter School in Abstract Analysis 2013section Set Theory & Topology 26th Jan — 2nd Feb 2013 The innovative structure of point sets in terms of the next point principle Sidorov O.V. Plekhanov Russian University of Economics Plekhanov Russia University of Economics

2. Motivation • Main notions of topology were extended from these of arithmetic and geometry • Geometrical notions and methods are clear, intuitive and natural as they are associated with reality. • Structure of point sets in particular sets on real line can be better analyzed in terms of Geometry. • Constructing real numbers in terms of arithmetic resulted in unsolvable problems. • For description of real space-time structure in present axiomatically generated real numbers are not sufficient Plekhanov Russia University of Economics

3. History • For a long time Geometry while it used Euclid’ methods was more advanced science than Arithmetic. And even Plato placed above the door of his Academia the words, "Let no one ignorant of geometry enter here“ • Natural, Rational, and Real numbers appeared due to needs of Geometry Plekhanov Russia University of Economics

4. History • However in the end of 19th century the correctness of foundations of Geometry was subjected to critique • Verification of the correctness was made by methods of Arythmetic • «The proof of the compatibility of the axioms in geometry is effected by constructing a suitable fieldof numbers, such that analogous relations between the numbers of this field correspond to thegeometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon berecognizable in the arithmetic of this field of numbers» • Hilbert D. Mathematical Problems Plekhanov Russia University of Economics

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

6. Real line • «We assume that you are familiar with the geometric interpretation of the real numbers as points on a line. …. Henceforth, we will use the terms real number system and real line synonymously and denote both by the symbol R; also, we will often refer to a real number as a point (on the real line) ». p.19 • Trench W.F. Introduction to Real Analysis (Prentice Hall), 2003 Plekhanov Russia University of Economics

7. Real line • «The reader is undoubtedly aware that the set of real numbers R may be represented as a horizontal line called the real line. Each point on the real line corresponds to a unique real number and each real number corresponds to a unique point on the line. We shall speak interchangeably of the set of real numbers R and the real line R» p.2 • Stancl D. L. Real analysis with point-set topology. MARCEL DEKKER, INC. 1987 Plekhanov Russia University of Economics

8. Real line • «A real lineis a linearly ordered field R satisfying the Bolzano Principle: every non-empty subset of R bounded from above has a supremum» p.39 • Bukovský L. The Structure of the Real Line, Springer Basel AG, 2011 Plekhanov Russia University of Economics

9. 3 Structure of Real line Dots are rationals, gaps are irrationals3 as a gap in the rationals Plekhanov Russia University of Economics

10. Intervals in R A closed interval in R An open interval in R Plekhanov Russia University of Economics

11. Summary • There is one-to-one correspondence between points on the real line and real numbers in R • The real line is a homogeneous and isotropic linear ordered infinite set of points • Structure of real line and intervals on it presupposes possibility of existence of next points for each given point • Properties of completeness and denseness of real numbers are incompatible with existence of next points for each given point Plekhanov Russia University of Economics

12. Main theory The Principle of next point «Any point of real lineintervalcontaining not less than 2 points has at least one next point» Example of real lineinterval is shown in the figure Proof. Consider the real line. It is a homogeneous and isotropic linear ordered infinite set of points, i.e actual infinity of points. Property of homogeneityimplies 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. Plekhanov Russia University of Economics

13. Main 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, and again 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. Plekhanov Russia University of Economics

14. Conclusion • Geometrical and Arithmetic interpretation of real numbers are not consistent • Another different model of real numbers can be constructed in terms of Geometry Plekhanov Russia University of Economics

15. Thank You Plekhanov Russia University of Economics