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0 (hodnocen0 x )

EB

EB

ONLINE

1st ed.

Prague : Karolinum Press, 2023

1 online zdroj (352 stran)

ISBN 9788024646633 (print)

ISBN 978-80-246-4664-0 (online ; pdf)

čeština

Cover -- Contents -- Editor’s Note -- Editor’s Introduction -- Part I Great Illusion of Twentieth Century Mathematics -- 1 Theological Foundations -- 1.1 Potential and Actual Infinity -- 1.1.1 Aurelius Augustinus (354-430) -- 1.1.2 Thomas Aquinas (1225-1274) -- 1.1.3 Giordano Bruno (1548-1600) -- 1.1.4 Galileo Galilei (1564-1654) -- 1.1.5 The Rejection of Actual Infinity -- 1.1.6 Infinitesimal Calculus -- 1.1.7 Number Magic -- 1.1.8 Jean le Rond d’Alembert (1717-1783) -- 1.2 The Disputation about Infinity in Baroque Prague -- 1.2.1 Rodrigo de Arriaga (1592-1667) -- 1.2.2 The Franciscan School -- 1.3 Bernard Bolzano (1781-1848) -- 1.3.1 Truth in Itself -- 1.3.2 The Paradox of the Infinite -- 1.3.3 Relational Structures on Infinite Multitudes -- 1.4 Georg Cantor (1845-1918) -- 1.4.1 Transfinite Ordinal Numbers -- 1.4.2 Actual Infinity -- 1.4.3 Rejection of Cantor’s Theory -- 2 Rise and Growth of Cantor’s Set Theory -- 2.1 Basic Notions -- 2.1.1 Relations and Functions -- 2.1.2 Orderings -- 2.1.3 Well-Orderings -- 2.2 Ordinal Numbers -- 2.3 Postulates of Cantor’s Set Theory -- 2.3.1 Cardinal Numbers -- 2.3.2 Postulate of the Powerset -- 2.3.3 Well-Ordering Postulate -- 2.3.4 Objections of French Mathematicians -- 2.4 Large Cardinalities -- 2.4.1 Initial Ordinal Numbers -- 2.4.2 Zorn’s Lemma -- 2.5 Developmental Influences -- 2.5.1 Colonisation of Infinitary Mathematics -- 2.5.2 Corpuses of Sets -- 2.5.3 Introduction of Mathematical Formalism in Set Theory -- 3 Explication of the Problem -- 3.1 Warnings -- 3.2 Two Further Emphatic Warnings -- 3.3 Ultrapower -- 3.4 There Exists No Set of All Natural Numbers -- 3.5 Unfortunate Consequences for All Infinitary Mathematics Based on Cantor’s Set Theory -- 4 Summit and Fall -- 4.1 Ultrafilters -- 4.2 Basic Language of Set Theory -- 4.3 Ultrapower Over a Covering Structure.

4.4 Ultraextension of the Domain of All Sets -- 4.5 Ultraextension Operator -- 4.6 Widening the Scope of Ultraextension Operator -- 4.7 Non-existence of the Set of All Natural Numbers -- 4.8 Extendable Domains of Sets -- 4.9 The Problem of Infinity -- Part II New Theory of Sets and Semisets -- 5 Basic Notions -- 5.1 Classes, Sets and Semisets -- 5.2 Horizon -- 5.3 Geometric Horizon -- 5.4 Finite Natural Numbers -- 6 Extension of Finite Natural Numbers -- 6.1 Natural Numbers within the Known Land of the Geometric Horizon -- 6.2 Axiom of Prolongation -- 6.3 Some Consequences of the Axiom of Prolongation -- 6.4 Revealed Classes -- 6.5 Forming Countable Classes -- 6.6 Cuts on Natural Numbers -- 7 Two Important Kinds of Classes -- 7.1 Motivation - Primarily Evident Phenomena -- 7.2 Mathematization… -- 7.3 Applications -- 7.4 Distortion of Natural Phenomena -- 8 Hierarchy of Descriptive Classes -- 8.1 Borel Classes -- 8.2 Analytic Classes -- 9 Topology -- 9.1 Motivation - Medial Look at Sets -- 9.2 Mathematization - Equivalence of Indiscernibility -- 9.3 Historical Intermezzo -- 9.4 The Nature of Topological Shapes -- 9.5 Applications: Invisible Topological Shapes -- 10 Synoptic Indiscernibility -- 10.1 Synoptic Symmetry of Indiscernibility -- 10.2 Geometric Equivalence of Indiscernibility -- 11 Further Non-traditional Motivations -- 11.1 Topological Misshapes -- 11.2 Imaginary Semisets -- 12 Search for Real Numbers -- 12.1 Liberation of the Domain of Real Numbers -- 12.2 Relation of Infinite Closeness on Rational Numbers in Known Land of Geometric Horizon -- 12.3 Real Numbers -- 12.4 Intermezzo About the Stars in the Sky -- 12.5 Interpretation of Real Numbers Corresponding to the First and Second phase in Interpreting Stars in the Sky -- 13 Classical Geometric World -- Part III Infinitesimal Calculus Reaffirmed -- Introduction.

14 Expansion of Ancient Geometric World -- 14.1 Ancient and Classical Geometric Worlds -- 14.2 Principles of Expansion -- 14.3 Infinitely Large Natural Numbers -- 14.4 Infinitely Large and Small Real Numbers -- 14.5 Infinite Closeness -- 14.6 Principles of Backward Projection -- 14.7 Arithmetic with Improper Numbers… -- 14.8 Further Fixed Notation for this Part -- 15 Sequences of Numbers -- 15.1 Binomial Numbers -- 15.2 Limits of Sequences -- 15.3 Euler’s Number -- 16 Continuity and Derivatives of Real Functions -- 16.1 Continuity of a Function at a Point -- 16.2 Derivative of a Function at a Point -- 16.3 Functions Continuous on a Closed Interval -- 16.4 Increasing and Decreasing Functions -- 16.5 Continuous Bijective Functions -- 16.6 Inverse Functions and Their Derivatives -- 16.7 Higher-Order Derivatives, Extrema and Points of Inflection -- 16.8 Limit of a Function at a Point -- 16.9 Taylor’s Expansion -- 17 Elementary Functions and Their Derivatives -- 17.1 Power Functions -- 17.2 Exponential Function -- 17.3 Logarithmic Function -- 17.4 Derivatives of Power, Exponential and Logarithmic Functions -- 17.5 Trigonometric Functions sin x, cos x and Their Derivatives -- 17.6 Trigonometric Functions tan x, cot x and Their Derivatives -- 17.7 Cyclometric Functions and Their Derivatives -- 18 Numerical Series -- 18.1 Convergence and Divergence -- 18.2 Series with Non-negative Terms -- 18.3 Convergence Criteria for Series with Positive Terms -- 18.4 Absolutely and Non-absolutely Convergent Series -- 19 Series of Functions -- 19.1 Taylor and Maclaurin Serie -- 19.2 Maclaurin Series of the Exponential Function -- 19.3 Maclaurin Series of Functions sin x, cos x -- 19.4 Powers of Complex Numbers -- 19.5 Maclaurin Series of the Function… -- 19.6 Maclaurin Series of the Function… -- 19.7 Binomial Series… -- 19.8 Series Expansion of the Function arctan x for….

19.9 Uniform Convergence -- Appendix to Part III - Translation Rules -- Part IV Making Real Numbers Discrete -- Introduction -- 20 Expansion of the Class Real of Real Numbers -- 20.1 Subsets of the Class Real -- 20.2 Third Principle of Expansion -- 21 Infinitesimal Arithmetics -- 21.1 Orders of Real Numbers -- 21.2 Near-Equality -- 22 Discretisation of the Ancient Geometric World -- 22.1 Grid -- 22.2 Fourth Principle of Expansion -- 22.3 Radius of Monads of a Full Almost-Uniform Grid -- Bibliography.

Popsáno podle dat dodaných poskytovatelem

001658819

Editor’s Note // Editor’s Introduction // Part I Great Illusion of Twentieth Century Mathematics // 1 Theological Foundations // 1.1 Potential and Actual Infinity // 1.1.1 Aurelius Augustinus (354-430) // 1.1.2 Thomas Aquinas (1225-1274) // 1.1.3 Giordano Bruno (1548-1600) // 1.1.4 Galileo Galilei (1564-1654) // 1.1.5 The Rejection of Actual Infinity // 1.1.6 Infinitesimal Calculus // 1.1.7 Number Magic // 1.1.8 Jean le Rond d’Alembert (1717-1783) // 1.2 The Disputation about Infinity in Baroque Prague // 1.2.1 Rodrigo de Arriaga (1592-1667) // 1.2.2 The Franciscan School // 1.3 Bernard Bolzano (1781-1848) // 1.3.1 Truth in Itself // 1.3.2 The Paradox of the Infinite // 1.3.3 Relational Structures on Infinite Multitudes // 1.4 Georg Cantor (1845-1918) // 1.4.1 Transfinite Ordinal Numbers // 1.4.2 Actual Infinity // 1.4.3 Rejection of Cantor’s Theory // 2 Rise and Growth of Cantor’s Set Theory // 2.1 Basic Notions // 2.1.1 Relations and Functions // 2.1.2 Orderings // 2.1.3 Well-Orderings // 2.2 Ordinal Numbers // 2.3 Postulates of Cantor’s Set Theory // 2.3.1 Cardinal Numbers // 2.3.2 Postulate of the Powerset // 2.3.3 Well-Ordering Postulate // 2.3.4 Objections of French Mathematicians // 2.4 Large Cardinalities // 2.4.1 Initial Ordinal Numbers // 2.4.2 Zorn’s Lemma // 2.5 Developmental Influences // 2.5.1 Colonisation of Infinitary Mathematics // 2.5.2 Corpuses of Sets // 2.5.3 Introduction of Mathematical Formalism in Set Theory // 3 Explication of the Problem // 3.1 Warnings // 3.2 Two Further Emphatic Warnings // 3.3 Ultrapower // 3.4 There Exists No Set of All Natural Numbers // 3.5 Unfortunate Consequences for All Infinitary Mathematics Based on Cantor’s Set Theory // 4 Summit and Fall // 4.1 Ultrafilters // 4.2 Basic Language of Set Theory // 4.3 Ultrapower Over a Covering Structure // 4.4 Ultraextension of the Domain of All Sets // 4.5 Ultraextension Operator //

4.6 Widening the Scope of Ultraextension Operator // 4.7 Non-existence of the Set of All Natural Numbers // 4.8 Extendable Domains of Sets // 4.9 The Problem of Infinity // Part II New Theory of Sets and Semisets // 5 Basic Notions // 5.1 Classes, Sets and Semisets // 5.2 Horizon // 5.3 Geometric Horizon // 5.4 Finite Natural Numbers // 6 Extension of Finite Natural Numbers // 6.1 Natural Numbers within the Known Land of the Geometric Horizon // 6.2 Axiom of Prolongation // 6.3 Some Consequences of the Axiom of Prolongation // 6.4 Revealed Classes // 6.5 Forming Countable Classes // 6.6 Cuts on Natural Numbers // 7 Two Important Kinds of Classes // 7.1 Motivation - Primarily Evident Phenomena // 7.2 Mathematization... // 7.3 Applications // 7.4 Distortion of Natural Phenomena // 8 Hierarchy of Descriptive Classes // 8.1 Borel Classes // 8.2 Analytic Classes // 9 Topology // 9.1 Motivation - Medial Look at Sets // 9.2 Mathematization - Equivalence of Indiscernibility // 9.3 Historical Intermezzo // 9.4 The Nature of Topological Shapes // 9.5 Applications: Invisible Topological Shapes // 10 Synoptic Indiscernibility // 10.1 Synoptic Symmetry of Indiscernibility // 10.2 Geometric Equivalence of Indiscernibility // 11 Further Non-traditional Motivations // 11.1 Topological Misshapes // 11.2 Imaginary Semisets // 12 Search for Real Numbers // 12.1 Liberation of the Domain of Real Numbers // 12.2 Relation of Infinite Closeness on Rational Numbers in Known Land of Geometric Horizon // 12.3 Real Numbers // 12.4 Intermezzo About the Stars in the Sky // 12.5 Interpretation of Real Numbers Corresponding to the First and Second phase in Interpreting Stars in the Sky // 13 Classical Geometric World // Part III Infinitesimal Calculus Reaffirmed // Introduction // 14 Expansion of Ancient Geometric World // 14.1 Ancient and Classical Geometric Worlds //

14.2 Principles of Expansion // 14.3 Infinitely Large Natural Numbers // 14.4 Infinitely Large and Small Real Numbers // 14.5 Infinite Closeness // 14.6 Principles of Backward Projection // 14.7 Arithmetic with Improper Numbers... // 14.8 Further Fixed Notation for this Part // 15 Sequences of Numbers // 15.1 Binomial Numbers // 15.2 Limits of Sequences // 15.3 Euler’s Number // 16 Continuity and Derivatives of Real Functions // 16.1 Continuity of a Function at a Point // 16.2 Derivative of a Function at a Point // 16.3 Functions Continuous on a Closed Interval // 16.4 Increasing and Decreasing Functions // 16.5 Continuous Bijective Functions // 16.6 Inverse Functions and Their Derivatives // 16.7 Higher-Order Derivatives, Extrema and Points of Inflection // 16.8 Limit of a Function at a Point // 16.9 Taylor’s Expansion // 17 Elementary Functions and Their Derivatives // 17.1 Power Functions // 17.2 Exponential Function // 17.3 Logarithmic Function // 17.4 Derivatives of Power, Exponential and Logarithmic Functions // 17.5 Trigonometric Functions sin x, cos x and Their Derivatives // 17.6 Trigonometric Functions tan x, cot x and Their Derivatives // 17.7 Cyclometric Functions and Their Derivatives // 18 Numerical Series // 18.1 Convergence and Divergence // 18.2 Series with Non-negative Terms // 18.3 Convergence Criteria for Series with Positive Terms // 18.4 Absolutely and Non-absolutely Convergent Series // 19 Series of Functions // 19.1 Taylor and Maclaurin Serie // 19.2 Maclaurin Series of the Exponential Function // 19.3 Maclaurin Series of Functions sin x, cos x // 19.4 Powers of Complex Numbers // 19.5 Maclaurin Series of the Function... // 19.6 Maclaurin Series of the Function... // 19.7 Binomial Series... // 19.8 Series Expansion of the Function arctan x for... //

19.9 Uniform Convergence // Appendix to Part III - Translation Rules // Part IV Making Real Numbers Discrete // Introduction // 20 Expansion of the Class Real of Real Numbers // 20.1 Subsets of the Class Real // 20.2 Third Principle of Expansion // 21 Infinitesimal Arithmetics // 21.1 Orders of Real Numbers // 21.2 Near-Equality // 22 Discretisation of the Ancient Geometric World // 22.1 Grid // 22.2 Fourth Principle of Expansion // 22.3 Radius of Monads of a Full Almost-Uniform Grid // Bibliography.

(MiAaPQ)EBC30130760

(Au-PeEL)EBL30130760

(OCoLC)1345587649

(MiAaPQ)ECB30130760