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1.1. Introduction
to the foundations 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. Binders in set theory

1.9. Axioms and proofs

1.10. Quantifiers

1.11. Second-order universal quantifiers

More philosophy:

1.A. Time in model theory

1.B. Truth undefinability

1.C. Introduction to incompleteness

1.D. Set theory as a unified framework

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. Binders in set theory

1.9. Axioms and proofs

1.10. Quantifiers

1.11. Second-order universal quantifiers

More philosophy:

1.A. Time in model theory

1.B. Truth undefinability

1.C. Introduction to incompleteness

1.D. Set theory as a unified framework

A notation change was done away from standards (see why) : from their definition in 2.6, the notation for direct images of sets by a graph

2.1. First axioms
of set theory

2.2. Set generation principle

2.3. Currying and tuples

2.4. Uniqueness quantifiers

2.5. Families, Boolean operators on sets

2.6. Graphs

2.7. Products and powerset

2.8. Injections, bijections

2.9. Properties of binary relations

2.10. Axiom of choice

Philosophical aspects :

2.A. Time in set theory

2.B. Interpretation of classes

2.C. Concepts of truth in mathematics

2.2. Set generation principle

2.3. Currying and tuples

2.4. Uniqueness quantifiers

2.5. Families, Boolean operators on sets

2.6. Graphs

2.7. Products and powerset

2.8. Injections, bijections

2.9. Properties of binary relations

2.10. Axiom of choice

Philosophical aspects :

2.A. Time in set theory

2.B. Interpretation of classes

2.C. Concepts of truth in mathematics

3.1. Galois
connections

3.2. Relational systems and concrete categories

3.3. Algebras

3.4. Special morphisms

3.5. Monoids and categories

3.6. Actions of monoids and categories

3.7. Invertibility and groups

3.8. Properties in categories

3.9. Initial and final objects

3.10. Products of systems

3.11. Basis

3.12. Composition of relations (new)

3.2. Relational systems and concrete categories

3.3. Algebras

3.4. Special morphisms

3.5. Monoids and categories

3.6. Actions of monoids and categories

3.7. Invertibility and groups

3.8. Properties in categories

3.9. Initial and final objects

3.10. Products of systems

3.11. Basis

3.12. Composition of relations (new)

4.1.
Algebraic terms

4.2. Quotient systems

4.3. Term algebras

4.4. Integers and recursion

4.5. Presburger Arithmetic

4.6. Finiteness (updated, June 2022)

4.7. Countability and Completeness

4.8. More recursion tools (draft)

4.9. Non-standard models of Arithmetic

4.10. Developing theories : definitions

4.11. Constructions

4.A. The Berry paradox

4.2. Quotient systems

4.3. Term algebras

4.4. Integers and recursion

4.5. Presburger Arithmetic

4.6. Finiteness (updated, June 2022)

4.7. Countability and Completeness

4.8. More recursion tools (draft)

4.9. Non-standard models of Arithmetic

4.10. Developing theories : definitions

4.11. Constructions

4.A. The Berry paradox

5.1. Second-order structures and invariants

5.2. Second-order logic

5.3. Well-foundedness

5.4. Ordinals and cardinals (draft)

5.5. Undecidability of the axiom of choice

5.6. Second-order arithmetic

5.7. The Incompleteness Theorem (draft)

More philosophical notes (uses Part 1
with philosophical
aspects + recursion) : 5.2. Second-order logic

5.3. Well-foundedness

5.4. Ordinals and cardinals (draft)

5.5. Undecidability of the axiom of choice

5.6. Second-order arithmetic

5.7. 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

6.1. Introduction to the foundations of geometry

6.2. Affine spaces

6.3. Duality

6.4. Vector spaces and barycenters

Beyond affine geometry

Euclidean geometry

6.2. Affine spaces

6.3. Duality

6.4. Vector spaces and barycenters

Beyond affine geometry

Euclidean geometry

Varieties

Polymorphisms and invariants

Relational clones

Abstract clones

Rings

(To be continued - see below drafts)

Polymorphisms and invariants

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

Index of special words, phrases and notations, with references

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

Affine geometry

Introduction to topology

Axiomatic expressions of Euclidean and Non-Euclidean geometries

Cardinals

An alternative to Zorn's Lemma

Introduction to inversive geometry

Affine geometry

Introduction to topology

Axiomatic expressions of Euclidean and Non-Euclidean geometries

Cardinals

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.