New Infinitary Mathematics

Petr Vopěnka (1935–2015) was a Czech mathematician and philosopher. In addition to teaching math and logic at Charles University, Jan Evangelista Purkyně University, and the University of West Bohemia, he also served as the Czech minister of education in the early 1990s. In mathematics, he is perhaps best known for establishing Vopěnka’s principle. Alena Vencovská is a Czech mathematician. Hana Moravcová is a Czech translator. Roland Andrew Letham translates from Czech. Václav Paris is a Czech translator.

• Editor’s Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiEditor’s Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiI Great Illusion of Twentieth Century Mathematics 211 Theological Foundations 251.1 Potential and Actual Infinity . . . . . . . . . . . . . . . . . . . . 251.1.1 Aurelius Augustinus (354–430) . . . . . . . . . . . . . . . 261.1.2 Thomas Aquinas (1225–1274) . . . . . . . . . . . . . . . . 271.1.3 Giordano Bruno (1548–1600) . . . . . . . . . . . . . . . . 291.1.4 Galileo Galilei (1564–1654) . . . . . . . . . . . . . . . . . 311.1.5 The Rejection of Actual Infinity . . . . . . . . . . . . . . 331.1.6 Infinitesimal Calculus . . . . . . . . . . . . . . . . . . . . 361.1.7 Number Magic . . . . . . . . . . . . . . . . . . . . . . . . 371.1.8 Jean le Rond d’Alembert (1717–1783) . . . . . . . . . . . 391.2 The Disputation about Infinity in Baroque Prague . . . . . . . . 411.2.1 Rodrigo de Arriaga (1592–1667) . . . . . . . . . . . . . . 411.2.2 The Franciscan School . . . . . . . . . . . . . . . . . . . . 471.3 Bernard Bolzano (1781–1848) . . . . . . . . . . . . . . . . . . . . 481.3.1 Truth in Itself . . . . . . . . . . . . . . . . . . . . . . . . . 481.3.2 The Paradox of the Infinite . . . . . . . . . . . . . . . . . 521.3.3 Relational Structures on Infinite Multitudes . . . . . . . 541.4 Georg Cantor (1845–1918) . . . . . . . . . . . . . . . . . . . . . . 561.4.1 Transfinite Ordinal Numbers . . . . . . . . . . . . . . . . 561.4.2 Actual Infinity . . . . . . . . . . . . . . . . . . . . . . . . 571.4.3 Rejection of Cantor’s Theory . . . . . . . . . . . . . . . . 582 Rise and Growth of Cantor’s Set Theory 672.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.1.1 Relations and Functions . . . . . . . . . . . . . . . . . . . 702.1.2 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.1.3 Well-Orderings . . . . . . . . . . . . . . . . . . . . . . . . 732.2 Ordinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.3 Postulates of Cantor’s Set Theory . . . . . . . . . . . . . . . . . 772.3.1 Cardinal Numbers . . . . . . . . . . . . . . . . . . . . . . 792.3.2 Postulate of the Powerset . . . . . . . . . . . . . . . . . . 812.3.3 Well-Ordering Postulate . . . . . . . . . . . . . . . . . . . 842.3.4 Objections of French Mathematicians . . . . . . . . . . . 862.4 Large Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . 892.4.1 Initial Ordinal Numbers . . . . . . . . . . . . . . . . . . . 892.4.2 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 912.5 Developmental Influences . . . . . . . . . . . . . . . . . . . . . . 922.5.1 Colonisation of Infinitary Mathematics . . . . . . . . . . . 922.5.2 Corpuses of Sets . . . . . . . . . . . . . . . . . . . . . . . 972.5.3 Introduction of Mathematical Formalismin Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . 983 Explication of the Problem 1033.1 Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2 Two Further Emphatic Warnings . . . . . . . . . . . . . . . . . . 1043.3 Ultrapower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.4 There Exists No Set of All Natural Numbers . . . . . . . . . . . 1073.5 Unfortunate Consequences for All Infinitary MathematicsBased on Cantor’s Set Theory . . . . . . . . . . . . . . . . . . . . 1094 Summit and Fall 1114.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Basic Language of Set Theory . . . . . . . . . . . . . . . . . . . . 1134.3 Ultrapower Over a Covering Structure . . . . . . . . . . . . . . . 1134.4 Ultraextension of the Domain of All Sets . . . . . . . . . . . . . . 1164.5 Ultraextension Operator . . . . . . . . . . . . . . . . . . . . . . . 1184.6 Widening the Scope of Ultraextension Operator . . . . . . . . . . 1194.7 Non-existence of the Set of All Natural Numbers . . . . . . . . . 1204.8 Extendable Domains of Sets . . . . . . . . . . . . . . . . . . . . 1214.9 The Problem of Infinity . . . . . . . . . . . . . . . . . . . . . . . 126II New Theory of Sets and Semisets 1295 Basic Notions 1355.1 Classes, Sets and Semisets . . . . . . . . . . . . . . . . . . . . . . 1355.2 Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3 Geometric Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.4 Finite Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . 1436 Extension of Finite Natural Numbers 1456.1 Natural Numbers within the Known Landof the Geometric Horizon . . . . . . . . . . . . . . . . . . . . . . 1456.2 Axiom of Prolongation . . . . . . . . . . . . . . . . . . . . . . . 1476.3 Some Consequences of the Axiom of Prolongation . . . . . . . . . 1486.4 Revealed Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.5 Forming Countable Classes . . . . . . . . . . . . . . . . . . . . . 1526.6 Cuts on Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 1577 Two Important Kinds of Classes 1597.1 Motivation – Primarily Evident Phenomena . . . . . . . . . . . . 1597.2 Mathematization: !-classes and ?-classes . . . . . . . . . . . . . 1627.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.4 Distortion of Natural Phenomena . . . . . . . . . . . . . . . . . 1698 Hierarchy of Descriptive Classes 1718.1 Borel Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2 Analytic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1749 Topology 1779.1 Motivation – Medial Look at Sets . . . . . . . . . . . . . . . . . 1779.2 Mathematization – Equivalence of Indiscernibility . . . . . . . . 1799.3 Historical Intermezzo . . . . . . . . . . . . . . . . . . . . . . . . . 1839.4 The Nature of Topological Shapes . . . . . . . . . . . . . . . . . 1849.5 Applications: Invisible Topological Shapes . . . . . . . . . . . . . 18610 Synoptic Indiscernibility 18910.1 Synoptic Symmetry of Indiscernibility . . . . . . . . . . . . . . . 18910.2 Geometric Equivalence of Indiscernibility . . . . . . . . . . . . . 19211 Further Non-traditional Motivations 19711.1 Topological Misshapes . . . . . . . . . . . . . . . . . . . . . . . . 19711.2 Imaginary Semisets . . . . . . . . . . . . . . . . . . . . . . . . . . 19812 Search for Real Numbers 20112.1 Liberation of the Domain of Real Numbers . . . . . . . . . . . . 20112.2 Relation of Infinite Closeness on Rational Numbersin the Known Land of Geometric Horizon . . . . . . . . . . . . . 20612.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20912.4 Intermezzo About the Stars in the Sky . . . . . . . . . . . . . . . 21112.5 Interpretation of Real Numbers Corresponding to theFirst and Second Phase in Interpreting Stars in the Sky . . . . . 21213 Classical Geometric World 215III Infinitesimal Calculus Rea_rmed 217Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21914 Expansion of Ancient Geometric World 22514.1 Ancient and Classical Geometric Worlds . . . . . . . . . . . . . . 22514.2 Principles of Expansion . . . . . . . . . . . . . . . . . . . . . . . 22614.3 Infinitely Large Natural Numbers . . . . . . . . . . . . . . . . . . 22714.4 Infinitely Large and Small Real Numbers . . . . . . . . . . . . . 22814.5 Infinite Closeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 23014.6 Principles of Backward Projection . . . . . . . . . . . . . . . . . 23114.7 Arithmetic with Improper Numbers 1, -1 . . . . . . . . . . . . 23314.8 Further Fixed Notation for this Part . . . . . . . . . . . . . . . . 23515 Sequences of Numbers 23715.1 Binomial Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 23715.2 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 23915.3 Euler’s Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24516 Continuity and Derivatives of Real Functions 24716.1 Continuity of a Function at a Point . . . . . . . . . . . . . . . . 24716.2 Derivative of a Function at a Point . . . . . . . . . . . . . . . . . 24816.3 Functions Continuous on a Closed Interval . . . . . . . . . . . . . 25116.4 Increasing and Decreasing Functions . . . . . . . . . . . . . . . . 25316.5 Continuous Bijective Functions . . . . . . . . . . . . . . . . . . . 25416.6 Inverse Functions and Their Derivatives . . . . . . . . . . . . . . 25516.7 Higher-Order Derivatives, Extrema and Points of Inflection . . . 25616.8 Limit of a Function at a Point . . . . . . . . . . . . . . . . . . . . 25916.9 Taylor’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 26417 Elementary Functions and Their Derivatives 26717.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26717.2 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . 27017.3 Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . 27217.4 Derivatives of Power, Exponential and Logarithmic Functions . . 27417.5 Trigonometric Functions sin x, cos x and Their Derivatives . . . . 27617.6 Trigonometric Functions tan x, cot x and Their Derivatives . . . 28117.7 Cyclometric Functions and Their Derivatives . . . . . . . . . . . 28318 Numerical Series 28718.1 Convergence and Divergence . . . . . . . . . . . . . . . . . . . . . 28718.2 Series with Non-negative Terms . . . . . . . . . . . . . . . . . . . 29318.3 Convergence Criteria for Series with Positive Terms . . . . . . . 29718.4 Absolutely and Non-absolutely Convergent Series . . . . . . . . . 30019 Series of Functions 30519.1 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . 30519.2 Maclaurin Series of the Exponential Function . . . . . . . . . . . 30619.3 Maclaurin Series of Functions sin x, cos x . . . . . . . . . . . . . . 30719.4 Powers of Complex Numbers . . . . . . . . . . . . . . . . . . . . 30819.5 Maclaurin Series of the Function log…………………. . . 31019.6 Maclaurin Series of the Function (1 + x)…………. . . . . . . . 31219.7 Binomial Series P"rn # xn for x = ±1 . . . . . . . . . . . . . . . . 31419.8 Series Expansion of the Function arctan x for .. . . . . . . . 31719.9 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . 320Appendix to Part III – Translation Rules 325IV Making Real Numbers Discrete 329Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33120 Expansion of the Class Real of Real Numbers 33320.1 Subsets of the Class Real . . . . . . . . . . . . . . . . . . . . . . 33320.2 Third Principle of Expansion . . . . . . . . . . . . . . . . . . . . 33421 Infinitesimal Arithmetics 33721.1 Orders of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 33721.2 Near-Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33822 Discretisation of the Ancient Geometric World 34122.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34122.2 Fourth Principle of Expansion . . . . . . . . . . . . . . . . . . . . 34322.3 Radius of Monads of a Full Almost-Uniform Grid . . . . . . . . . 344Bibliography 347

"An interesting historical introduction to the concept of infinity as a stimulating topic for theologists, philosophers and researchers of natural science throughout the centuries."