Table of Contents

Table of Contents

"Introduction to Mathematical Logics"

INTRODUCTION

BASIC MATHEMATICAL LOGIC

ADVANCED MATHEMATICAL LOGIC

NUMBER THEORY

"Introduction to Mathematical Logics"

"Introduction to Mathematical Logics"

SIMONE MALACRIDA

In this book, all facets of mathematical logic are presented such as:

symbology, principles and properties of elementary logic

boolean logic

order theory and axiomatic systems

axiomatic set theory and Godel's theorems

logical paradoxes and logical antinomies

descriptive and fuzzy logics

number theory and modular arithmetic

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Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – BASIC MATHEMATICAL LOGIC

Introduction

Symbology

Principles

Property

Boolean logic

Applications of logic: proof of theorems

Applications of Boolean logic: electronic calculators

Insight: syllogism and mathematical logic

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II – ADVANCED MATHEMATICAL LOGIC

Order theory

Robinson and Peano arithmetic

Axiomatic systems

Axiomatic set theory

Godel's theorems

Paradoxes and antinomies

Other logical systems

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III – NUMBER THEORY

Definitions

Modular arithmetic

INTRODUCTION

INTRODUCTION

This book presents all the topics concerning mathematical logic which is the basic tool for understanding any subsequent scientific knowledge.

First, basic knowledge is introduced, such as the use of logical connectors, logical definitions and terminology, as well as Boolean logic and logical principles already used by the ancients.

Subsequently, the purely modern and contemporary part of logic will be exposed, such as the theory of orders and the axiomatic theory of sets, giving ample space to axiomatic systems and the fundamental theorems of Godel, one of the cornerstones of twentieth-century knowledge.

Logical paradoxes and antinomies are a prerequisite for overcoming normal mathematical logic, towards much more open schemes, such as that of fuzzy logic.

Finally, number theory and modular arithmetic are a testing ground for logic itself, still having to prove many conjectures.

The cut of the book is deliberately technical and concise, just to get lost in frills and to give the reader a clear picture of a discipline halfway between mathematics and philosophy.

The first chapter can be understood through high school knowledge, while the next two certainly require university notions.

I

BASIC MATHEMATICAL LOGIC

BASIC MATHEMATICAL LOGIC

Introduction

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Mathematical logic deals with the coding, in mathematical terms, of intuitive concepts related to human reasoning.

It is the starting point for any mathematical learning process and, therefore, it makes complete sense to expose the elementary rules of this logic at the beginning of the whole discourse.

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We define an axiom as a statement assumed to be true because it is considered self-evident or because it is the starting point of a theory.

Logical axioms are satisfied by any logical structure and are divided into tautologies (true statements by definition devoid of new informative value) or axioms considered true regardless, unable to demonstrate their universal validity.

Non-logical axioms are never tautologies and are called postulates .

Both axioms and postulates are unprovable.

Generally, the axioms that found and start a theory are called principles .

A theorem , on the other hand, is a proposition which, starting from initial conditions (called hypotheses ) reaches conclusions (called theses ) through a logical procedure called demonstration .

Theorems are, therefore, provable by definition.

Other provable statements are the lemmas which usually precede and give the basis of a theorem and the corollaries which, instead, are consequent to the demonstration of a given theorem.

A conjecture , on the other hand, is a proposition believed to be true thanks to general considerations, intuition and common sense, but not yet demonstrated in the form of a theorem.

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Symbology

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Mathematical logic causes symbols to intervene which will then return in all the individual fields of mathematics. These symbols are varied and belong to different categories.

The equality between two mathematical elements is indicated with the symbol of , if instead these elements are different from each other the symbol of inequality is given by .

In the geometric field it is also useful to introduce the concept of congruence , indicated in this way and of similarity .

proportionality can also be defined , denoted by .

In many cases mathematical and geometric concepts must be defined, the definition symbol is this .

Finally, the negation is given by a bar above the logical concept.

Then there are quantitative

Impressum

Verlag: BookRix GmbH & Co. KG

Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3940-0

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