New Infinitary Mathematics.

Format:

Book

Author/Creator:

Vopěnka, Petr.

Language:

English

Subjects (All):

Mathematics--Philosophy.

Mathematics.

Physical Description:

1 online resource (352 pages)

Edition:

1st ed.

Place of Publication:

Prague : Karolinum Press, 2023.

Summary:

Pro matematiku dvacátého století je příznačné, že její hlavní proud zkoumající a zároveň aplikující nekonečno, byť v bizarních ideálních světech, je založen na klasické Cantorově teorii nekonečných množin. Ta sama se pak opírá o problematický předpoklad existence množiny všech přirozených čísel, jehož jediné – a to navíc teologické – odůvodnění bývá zamlčováno a vytlačováno do kolektivního nevědomí. Kniha nejprve zkoumá teologické základy, z nichž klasická teorie množin vznikla a na nichž se rozvíjela. Autor varuje před nebezpečími skrytými v konstrukci teorie množin, která lze vysledovat v pracích některých významných matematiků, jakož i v jeho vlastních pracích. Poté předkládá argument o absurditě předpokladu existence množiny všech přirozených čísel. Autorem budovaná nová infinitní matematika není však jen pouhou negací současných názorů a předpokladů. Naopak, jeho teorie je vedena opatrnou snahou o nová překračování obzoru ohraničujícího antický geometrický svět a předmnožinovou matematiku a snaží se o těsnější korespondenci s přirozeným reálným světem kolem nás. Druhá polovina textu je věnována rehabilitaci nekonečně malých veličin i jejich zásadní role v matematické analýze.

Contents:

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.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Vopěnka, Petr New Infinitary Mathematics

ISBN:

9788024646640

OCLC:

1345587649