Other languages : FR − RU − TR − 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

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 ; 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 ; ordered sets

2.8. Canonical bijections

2.9. Equivalence relations and partitions

2.10. Axiom of choice

2.11. Galois connection

3.1. Relational systems and concrete categories

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7. Categories

3.8. Initial and final objects (expanded, March 2018)

3.9. Algebraic terms

3.10. Term algebras (still incomplete)

3.11. Integers and recursion

3.12. Presburger Arithmetic

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7. Categories

3.8. Initial and final objects (expanded, March 2018)

3.9. Algebraic terms

3.10. Term algebras (still incomplete)

3.11. Integers and recursion

3.12. Presburger Arithmetic

4.1. Finiteness and countability (draft)

4.2. The Completeness Theorem

4.3. Non-standard models of Arithmetic

4.4. How theories develop

4.5. Second-order logic (last updated, dec. 2017)

4.6. Well-foundedness

4.7. Ordinals and cardinals (draft)

4.8. Undecidability of the axiom of choice

4.9. Second-order arithmetic

4.10. The Incompleteness Theorem (draft)

More philosophical notes (uses Part 1
with philosophical
aspects + recursion) : 4.2. The Completeness Theorem

4.3. Non-standard models of Arithmetic

4.4. How theories develop

4.5. Second-order logic (last updated, dec. 2017)

4.6. Well-foundedness

4.7. Ordinals and cardinals (draft)

4.8. Undecidability of the axiom of choice

4.9. Second-order arithmetic

4.10. The Incompleteness Theorem (draft)

Gödelian arguments against mechanism : what was wrong and how to do instead

Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system

Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system

5.1. Introduction to the foundations of geometry

5.2. First-order invariants in concrete categories

5.3. Second-order invariants

5.4. Affine spaces

5.5. Vector spaces and barycenters

Beyond affine geometry

Euclidean geometry

5.2. First-order invariants in concrete categories

5.3. Second-order invariants

5.4. Affine spaces

5.5. Vector spaces and barycenters

Beyond affine geometry

Euclidean geometry

Products of
systems

Varieties

Polymorphisms, invariants and clones of operations

Relational clones

Abstract clones

Rings

(To be continued - see below drafts)

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))

Resolution of cubic equations

Outer automorphisms of S_{6}

Resolution of cubic equations

Outer automorphisms of S

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.