Distributed for Karolinum Press, Charles University
New Infinitary Mathematics
A rethinking of Cantor and infinitary mathematics by the creator of Vopěnka'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.
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
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