Other languages : FR − RU − ES

.

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

Time
in model theory (updated, dec. 2016)

Time in set theory

Interpretation of classes

Concepts of truth in mathematics

Time in set theory

Interpretation of classes

Concepts of truth in mathematics

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 (updated, jan. 2017)

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 (updated, jan. 2017)

2.11. Galois connection

3.1. Morphisms
of relational systems and concrete categories (updated, feb. 2017)

3.2. Special morphisms (updated, dec. 2016)

3.3. Algebras (updated, dec. 2016)

3.4. Algebraic terms and term algebras (updated, dec. 2016)

3.5. Integers and recursion (updated, jan. 2017)

3.6. Arithmetic with addition

3.2. Special morphisms (updated, dec. 2016)

3.3. Algebras (updated, dec. 2016)

3.4. Algebraic terms and term algebras (updated, dec. 2016)

3.5. Integers and recursion (updated, jan. 2017)

3.6. Arithmetic with addition

4.1. Finiteness and countability (new, jan. 2017)

4.2. The Completeness Theorem (updated, jan. 2017)

4.3. Infinity and the axiom of choice (new, jan. 2017)

4.4. Non-standard models of Arithmetic (updated, jan. 2017)

4.5. How theories develop (updated, feb. 2017)

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

More philosophical notes : 4.2. The Completeness Theorem (updated, jan. 2017)

4.3. Infinity and the axiom of choice (new, jan. 2017)

4.4. Non-standard models of Arithmetic (updated, jan. 2017)

4.5. How theories develop (updated, feb. 2017)

4.6. 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)

Monoids
and actions

Groups, automorphisms and invariants

Second-order logic

Introduction to the foundations of geometry

Groups, automorphisms and invariants

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 (Sep. 2012 - before that,
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.