Mathematical Logic On Numbers Sets Structures and Symmetry 1st Edition by Roman Kossak 3030073319 9783319972978

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ISBN-10 :   3030073319 

ISBN-13 :  9783319972978

Author:  Roman Kossak 

Presents an introduction to of formal mathematical logic and set theory

Presents simple yet nontrivial results in modern model theory

Provides introductory remarks to all results, including a historical background

 

Mathematical Logic On Numbers Sets Structures and Symmetry 1st Table of contents:

Part I Logic, Sets, and Numbers

1 First-Order Logic..
1.1 What We Talk About When We Talk About Numbers..
How to Choose a Vocabulary?
1.2 Symbolic Logic..
1.2.1 Trivial Structures

2 Logical Seeing
2.1 Finite Graphs
2.2 Symmetry
2.3 Types and Fixed Points..
2.4 Seeing Numbers

3 What Is a Number?.
3.1 How Natural Are the Natural Numbers?.
3.1.1 Arithmetic Operations and the Decimal System.
3.1.2 How Many Numbers Are There?
3.1.3 Zero
3.1.4 The Set of Natural Numbers

4 Seeing the Number Structures
4.1 What Is the Structure of the Natural Numbers?.
4.1.1 Sets and Set Notation..
4.1.2 Language of Formal Arithmetic
4.1.3 Linearly Ordered Sets
4.1.4 The Ordering of the Natural Numbers.
4.2 The Arithmetic Structure of the Natural Numbers.
4.3 The Arithmetic Structure of the Integers..
4.3.1 Natural Numbers Are Integers, Naturally
4.4 Fractions!

5 Points, Lines, and the Structure of R
5.1 Density of Rational Numbers
5.2 What Are Real Numbers, Really?
5.3 Dedekind Cuts
5.3.1 Dedekind Complete Orderings.
5.3.2 Summary
5.4 Dangerous Consequences
5.5 Infinite Decimals
6 Set Theory
6.1 What to Assume About Infinite Sets?

Part II Relations, Structures, Geometry

7 Relations
7.1 Ordered Pairs
7.2 Cartesian Products.
7.3 What Is a Relation on a Set?..
7.4 Definability: New Relations from Old
7.5 How Many Different Structures Are There?
7.5.1 A Very Small Complex Structure

8 Definable Elements and Constants
8.1 Definable Elements
8.2 Databases, Oracles, and the Theory of a Structure
8.3 Defining Real Numbers
8.4 Definability With and Without Parameters.

9 Minimal and Order-Minimal Structures
9.1 Types, Symmetries, and Minimal Structures.
9.2 Trivial Structures
9.3 The Ordering of the Natural Numbers
9.4 The Ordering of the Integers.
9.5 The Additive Structure of the Integers
9.6 The Ordering of the Rational Numbers

10 Geometry of Definable Sets

 

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Tags: Mathematical Logic, Numbers Sets, Structures, Symmetry, Roman Kossak