A rethinking of Cantor and infinitary mathematics by the creator of Vopenka's principle.   The dominant current of twentieth-century mathematics relies on Georg Cantor’s classical theory of infinite sets, which in turn relies on the assumption of the existence of the set of all natural numbers, the only justification for which—a theological justification—is usually concealed and pushed into the background. This book surveys the theological background, emergence, and development of classical set theory, warning us about the dangers implicit in the construction of set theory, and presents an argument about the absurdity of the assumption of the existence of the set of all natural numbers. It instead proposes and develops a new infinitary mathematics driven by a cautious effort to transcend the horizon bounding the ancient geometric world and mathematics prior to set theory, while allowing mathematics to correspond more closely to the real world surrounding us. Finally, it discusses real numbers and demonstrates how, within a new infinitary mathematics, calculus can be rehabilitated in its original form employing infinitesimals.

Table of Contents:
Editor’s Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Editor’s Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii I Great Illusion of Twentieth Century Mathematics 21 1 Theological Foundations 25 1.1 Potential and Actual Infinity . . . . . . . . . . . . . . . . . . . . 25 1.1.1 Aurelius Augustinus (354–430) . . . . . . . . . . . . . . . 26 1.1.2 Thomas Aquinas (1225–1274) . . . . . . . . . . . . . . . . 27 1.1.3 Giordano Bruno (1548–1600) . . . . . . . . . . . . . . . . 29 1.1.4 Galileo Galilei (1564–1654) . . . . . . . . . . . . . . . . . 31 1.1.5 The Rejection of Actual Infinity . . . . . . . . . . . . . . 33 1.1.6 Infinitesimal Calculus . . . . . . . . . . . . . . . . . . . . 36 1.1.7 Number Magic . . . . . . . . . . . . . . . . . . . . . . . . 37 1.1.8 Jean le Rond d’Alembert (1717–1783) . . . . . . . . . . . 39 1.2 The Disputation about Infinity in Baroque Prague . . . . . . . . 41 1.2.1 Rodrigo de Arriaga (1592–1667) . . . . . . . . . . . . . . 41 1.2.2 The Franciscan School . . . . . . . . . . . . . . . . . . . . 47 1.3 Bernard Bolzano (1781–1848) . . . . . . . . . . . . . . . . . . . . 48 1.3.1 Truth in Itself . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.3.2 The Paradox of the Infinite . . . . . . . . . . . . . . . . . 52 1.3.3 Relational Structures on Infinite Multitudes . . . . . . . 54 1.4 Georg Cantor (1845–1918) . . . . . . . . . . . . . . . . . . . . . . 56 1.4.1 Transfinite Ordinal Numbers . . . . . . . . . . . . . . . . 56 1.4.2 Actual Infinity . . . . . . . . . . . . . . . . . . . . . . . . 57 1.4.3 Rejection of Cantor’s Theory . . . . . . . . . . . . . . . . 58 2 Rise and Growth of Cantor’s Set Theory 67 2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1.1 Relations and Functions . . . . . . . . . . . . . . . . . . . 70 2.1.2 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.1.3 Well-Orderings . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2 Ordinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.3 Postulates of Cantor’s Set Theory . . . . . . . . . . . . . . . . . 77 2.3.1 Cardinal Numbers . . . . . . . . . . . . . . . . . . . . . . 79 2.3.2 Postulate of the Powerset . . . . . . . . . . . . . . . . . . 81 2.3.3 Well-Ordering Postulate . . . . . . . . . . . . . . . . . . . 84 2.3.4 Objections of French Mathematicians . . . . . . . . . . . 86 2.4 Large Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.4.1 Initial Ordinal Numbers . . . . . . . . . . . . . . . . . . . 89 2.4.2 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 91 2.5 Developmental Influences . . . . . . . . . . . . . . . . . . . . . . 92 2.5.1 Colonisation of Infinitary Mathematics . . . . . . . . . . . 92 2.5.2 Corpuses of Sets . . . . . . . . . . . . . . . . . . . . . . . 97 2.5.3 Introduction of Mathematical Formalism in Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . 98 3 Explication of the Problem 103 3.1 Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2 Two Further Emphatic Warnings . . . . . . . . . . . . . . . . . . 104 3.3 Ultrapower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.4 There Exists No Set of All Natural Numbers . . . . . . . . . . . 107 3.5 Unfortunate Consequences for All Infinitary Mathematics Based on Cantor’s Set Theory . . . . . . . . . . . . . . . . . . . . 109 4 Summit and Fall 111 4.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Basic Language of Set Theory . . . . . . . . . . . . . . . . . . . . 113 4.3 Ultrapower Over a Covering Structure . . . . . . . . . . . . . . . 113 4.4 Ultraextension of the Domain of All Sets . . . . . . . . . . . . . . 116 4.5 Ultraextension Operator . . . . . . . . . . . . . . . . . . . . . . . 118 4.6 Widening the Scope of Ultraextension Operator . . . . . . . . . . 119 4.7 Non-existence of the Set of All Natural Numbers . . . . . . . . . 120 4.8 Extendable Domains of Sets . . . . . . . . . . . . . . . . . . . . 121 4.9 The Problem of Infinity . . . . . . . . . . . . . . . . . . . . . . . 126 II New Theory of Sets and Semisets 129 5 Basic Notions 135 5.1 Classes, Sets and Semisets . . . . . . . . . . . . . . . . . . . . . . 135 5.2 Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Geometric Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Finite Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . 143 6 Extension of Finite Natural Numbers 145 6.1 Natural Numbers within the Known Land of the Geometric Horizon . . . . . . . . . . . . . . . . . . . . . . 145 6.2 Axiom of Prolongation . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Some Consequences of the Axiom of Prolongation . . . . . . . . . 148 6.4 Revealed Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.5 Forming Countable Classes . . . . . . . . . . . . . . . . . . . . . 152 6.6 Cuts on Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 157 7 Two Important Kinds of Classes 159 7.1 Motivation – Primarily Evident Phenomena . . . . . . . . . . . . 159 7.2 Mathematization: !-classes and ?-classes . . . . . . . . . . . . . 162 7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.4 Distortion of Natural Phenomena . . . . . . . . . . . . . . . . . 169 8 Hierarchy of Descriptive Classes 171 8.1 Borel Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.2 Analytic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9 Topology 177 9.1 Motivation – Medial Look at Sets . . . . . . . . . . . . . . . . . 177 9.2 Mathematization – Equivalence of Indiscernibility . . . . . . . . 179 9.3 Historical Intermezzo . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.4 The Nature of Topological Shapes . . . . . . . . . . . . . . . . . 184 9.5 Applications: Invisible Topological Shapes . . . . . . . . . . . . . 186 10 Synoptic Indiscernibility 189 10.1 Synoptic Symmetry of Indiscernibility . . . . . . . . . . . . . . . 189 10.2 Geometric Equivalence of Indiscernibility . . . . . . . . . . . . . 192 11 Further Non-traditional Motivations 197 11.1 Topological Misshapes . . . . . . . . . . . . . . . . . . . . . . . . 197 11.2 Imaginary Semisets . . . . . . . . . . . . . . . . . . . . . . . . . . 198 12 Search for Real Numbers 201 12.1 Liberation of the Domain of Real Numbers . . . . . . . . . . . . 201 12.2 Relation of Infinite Closeness on Rational Numbers in the Known Land of Geometric Horizon . . . . . . . . . . . . . 206 12.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 12.4 Intermezzo About the Stars in the Sky . . . . . . . . . . . . . . . 211 12.5 Interpretation of Real Numbers Corresponding to the First and Second Phase in Interpreting Stars in the Sky . . . . . 212 13 Classical Geometric World 215 III Infinitesimal Calculus Rea_rmed 217 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 14 Expansion of Ancient Geometric World 225 14.1 Ancient and Classical Geometric Worlds . . . . . . . . . . . . . . 225 14.2 Principles of Expansion . . . . . . . . . . . . . . . . . . . . . . . 226 14.3 Infinitely Large Natural Numbers . . . . . . . . . . . . . . . . . . 227 14.4 Infinitely Large and Small Real Numbers . . . . . . . . . . . . . 228 14.5 Infinite Closeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 14.6 Principles of Backward Projection . . . . . . . . . . . . . . . . . 231 14.7 Arithmetic with Improper Numbers 1, -1 . . . . . . . . . . . . 233 14.8 Further Fixed Notation for this Part . . . . . . . . . . . . . . . . 235 15 Sequences of Numbers 237 15.1 Binomial Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 237 15.2 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 239 15.3 Euler’s Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 16 Continuity and Derivatives of Real Functions 247 16.1 Continuity of a Function at a Point . . . . . . . . . . . . . . . . 247 16.2 Derivative of a Function at a Point . . . . . . . . . . . . . . . . . 248 16.3 Functions Continuous on a Closed Interval . . . . . . . . . . . . . 251 16.4 Increasing and Decreasing Functions . . . . . . . . . . . . . . . . 253 16.5 Continuous Bijective Functions . . . . . . . . . . . . . . . . . . . 254 16.6 Inverse Functions and Their Derivatives . . . . . . . . . . . . . . 255 16.7 Higher-Order Derivatives, Extrema and Points of Inflection . . . 256 16.8 Limit of a Function at a Point . . . . . . . . . . . . . . . . . . . . 259 16.9 Taylor’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 264 17 Elementary Functions and Their Derivatives 267 17.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 17.2 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . 270 17.3 Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . 272 17.4 Derivatives of Power, Exponential and Logarithmic Functions . . 274 17.5 Trigonometric Functions sin x, cos x and Their Derivatives . . . . 276 17.6 Trigonometric Functions tan x, cot x and Their Derivatives . . . 281 17.7 Cyclometric Functions and Their Derivatives . . . . . . . . . . . 283 18 Numerical Series 287 18.1 Convergence and Divergence . . . . . . . . . . . . . . . . . . . . . 287 18.2 Series with Non-negative Terms . . . . . . . . . . . . . . . . . . . 293 18.3 Convergence Criteria for Series with Positive Terms . . . . . . . 297 18.4 Absolutely and Non-absolutely Convergent Series . . . . . . . . . 300 19 Series of Functions 305 19.1 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . 305 19.2 Maclaurin Series of the Exponential Function . . . . . . . . . . . 306 19.3 Maclaurin Series of Functions sin x, cos x . . . . . . . . . . . . . . 307 19.4 Powers of Complex Numbers . . . . . . . . . . . . . . . . . . . . 308 19.5 Maclaurin Series of the Function log…………………. . . 310 19.6 Maclaurin Series of the Function (1 + x)…………. . . . . . . . 312 19.7 Binomial Series P"rn # xn for x = ±1 . . . . . . . . . . . . . . . . 314 19.8 Series Expansion of the Function arctan x for .. . . . . . . . 317 19.9 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . 320 Appendix to Part III – Translation Rules 325 IV Making Real Numbers Discrete 329 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 20 Expansion of the Class Real of Real Numbers 333 20.1 Subsets of the Class Real . . . . . . . . . . . . . . . . . . . . . . 333 20.2 Third Principle of Expansion . . . . . . . . . . . . . . . . . . . . 334 21 Infinitesimal Arithmetics 337 21.1 Orders of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 337 21.2 Near-Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 22 Discretisation of the Ancient Geometric World 341 22.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 22.2 Fourth Principle of Expansion . . . . . . . . . . . . . . . . . . . . 343 22.3 Radius of Monads of a Full Almost-Uniform Grid . . . . . . . . . 344 Bibliography 347