Other languages : FR − RU − ES

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1.1. Introduction
to the foundation of mathematics

1.2. Variables, sets, functions and operations

1.3. Form of theories: notions, objects, meta-objects

1.4. Structures of mathematical systems

1.5. Expressions and definable structures

1.6. Logical connectives

1.7. Classes in set theory

1.8. Bound variables in set theory

1.9. Quantifiers

1.10. Formalization of set theory

1.11. Set generation principle

1.2. Variables, sets, functions and operations

1.3. Form of theories: notions, objects, meta-objects

1.4. Structures of mathematical systems

1.5. Expressions and definable structures

1.6. Logical connectives

1.7. Classes in set theory

1.8. Bound variables in set theory

1.9. Quantifiers

1.10. Formalization of set theory

1.11. Set generation principle

2.1. Tuples, families

2.2. Boolean operators on families of sets

2.3. Products, graphs and composition

2.4. Uniqueness quantifiers, functional graphs

2.5. The powerset axiom

2.6. Injectivity and inversion

2.7. Properties of binary relations on a set ; ordered sets

2.8. Canonical bijections

2.9. Equivalence relations and partitions

2.10. Axiom of choice

2.11. Galois connection

2.2. Boolean operators on families of sets

2.3. Products, graphs and composition

2.4. Uniqueness quantifiers, functional graphs

2.5. The powerset axiom

2.6. Injectivity and inversion

2.7. Properties of binary relations on a set ; ordered sets

2.8. Canonical bijections

2.9. Equivalence relations and partitions

2.10. Axiom of choice

2.11. Galois connection

3.1. Morphisms of relational systems and concrete categories

3.2. Special morphisms

3.3. Algebras

3.4. Algebraic terms and term algebras

3.5. Integers and recursion

3.6. The Completeness Theorem (updated, June 2016)

3.7. How theories develop (updated, June 2016)

3.8. Arithmetic with addition

3.9. Non-standard models of Arithmetic (updated, August 2016)

3.10. The Incompleteness Theorem (updated, Sept. 2016 ; more improvement still possible)

More philosophical notes : 3.2. Special morphisms

3.3. Algebras

3.4. Algebraic terms and term algebras

3.5. Integers and recursion

3.6. The Completeness Theorem (updated, June 2016)

3.7. How theories develop (updated, June 2016)

3.8. Arithmetic with addition

3.9. Non-standard models of Arithmetic (updated, August 2016)

3.10. The Incompleteness Theorem (updated, Sept. 2016 ; more improvement still possible)

Philosophical
proof of consistency of the Zermelo-Fraenkel axiomatic system
(uses Part 1 with
philosophical aspects + recursion)

(List of texts on algebra)

Monoids
and groups

The Galois connection between structures and permutations (Automorphisms, Invariants)

Actions of monoids and groups

Second-order logic

Introduction to the foundations of geometry

The Galois connection between structures and permutations (Automorphisms, Invariants)

Actions of monoids and groups

Second-order logic

Introduction to the foundations of geometry

What is geometry

Structures and permutations in the plane

Affine geometry

Beyond affine geometry

Euclidean geometry

The completeness of first-order geometry

Structures and permutations in the plane

Affine geometry

Beyond affine geometry

Euclidean geometry

The completeness of first-order geometry

Second-order
arithmetic

General formalization tools (draft)

Products of systems

Varieties

Polymorphisms, invariants and clones of operations

Relational clones

Abstract clones

Rings

(To be continued - see below drafts)

General formalization tools (draft)

Products of systems

Varieties

Polymorphisms, invariants and clones of operations

Relational clones

Abstract clones

Rings

(To be continued - see below drafts)

Monotone Galois connections (adjunctions)

Upper and lower bounds, infimum and supremum

Complete lattices

Fixed point theorem

Transport of closure

Preorder generated by a relation

Finite sets

Generated equivalence relations, and more

Well-founded relations

Dimensional analysis : Quantities and real numbers - incomplete draft text of a video lecture I wish to make on 1-dimensional geometry

Introduction to inversive geometry

Duality systems and theories

Affine geometry

Introduction to topology

Vector spaces in duality

Axiomatic expressions of Euclidean and Non-Euclidean geometries

Cardinals

Well-orderings and ordinals (with an alternative to Zorn's Lemma).

Introduction to inversive geometry

Duality systems and theories

Affine geometry

Introduction to topology

Vector spaces in duality

Axiomatic expressions of Euclidean and Non-Euclidean geometries

Cardinals

Well-orderings and ordinals (with an alternative to Zorn's Lemma).

Pythagorean
triples (triples of integers (a,b,c) forming the sides of a
right triangle, such as (3,4,5))

I wrote large parts of the Wikipedia article on Foundations
of mathematics (September 2012 - because until then,
other authors focused on the more professional and technical
article Mathematical
logic instead; the Foundations of mathematics article is
more introductory, historical and philosophical) and improved
the one on the completeness
theorem.