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**1. First
foundations of mathematics** (detailed list of
sections) - pdf
version (13 + 6 pages, updated on July, 2014 - full text in 1 html
page).

1.1. Introduction
to the foundation of mathematics

1.2. Variables, sets, functions and operations

1.3. Structure of theories: notions, objects and meta-objects

1.4. Formalizing types and structures

1.5. Expressions and definable structures

1.6. Connectives

1.7. Classes in set theory

1.8. Bound variables in set theory

1.9. Quantifiers

1.10. First axioms of set theory

1.11. Set generation principle

1.2. Variables, sets, functions and operations

1.3. Structure of theories: notions, objects and meta-objects

1.4. Formalizing types and structures

1.5. Expressions and definable structures

1.6. Connectives

1.7. Classes in set theory

1.8. Bound variables in set theory

1.9. Quantifiers

1.10. First axioms of set theory

1.11. Set generation principle

**2. Set theory
(continued) (11
pdf pages)**

2.1. Tuples, families

2.2. Boolean operators on families of sets

2.3. Products, graphs and composition

2.4. Uniqueness quantifier

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 connections

2.2. Boolean operators on families of sets

2.3. Products, graphs and composition

2.4. Uniqueness quantifier

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 connections

3.1. What is a mathematical theoryMore philosophical notes : About the powerset axiom

3.2. How mathematical theories develop

3.3. Notion of Algebra

3.4. Relational systems and their morphisms

3.5. The Galois connection between (invariant) structures and permutations (automorphisms)

3.6. Second-order theories

3.7. Formalizations of Arithmetic

3.8. Non-standard models of Arithmetic

3.9. The Incompleteness Theorem

(List of texts on algebra)

The following text makes rigorously no use of texts 3 and 4, but only uses text 1 (without complements) and 2. Its position has been moved from 3 to 5 for pedagogical reasons (higher difficulty level while the above texts 3 and 4 are more directly interesting).Introduction to the foundations of geometry

What is geometryProducts of relational systems

Structures and permutations in the plane

Affine geometry

Beyond affine geometry

Euclidean geometry

The completeness of first-order geometry

Truth of formulas in productsPolymorphisms and invariants

Morphisms into products

Products of algebras

The Galois connection Inv-Pol between sets of operations and relationsDuality theories (to be completed)

The Galois connection Pol-Pol between sets of operations

Morphisms of duality systems

Duality theories with structures

Algebraic duality theories[This section 4 has only been well worked on to this point; the below is a draft, to be reworked later]Affine geometry

Introduction to inversive geometry

Introduction to topology

Monoids, groups and actions

From transformation monoids to abstract monoidsVector spaces in duality

Monoids, theories and morphisms

Groups

Dimensional analysis

Axiomatic expressions of Euclidean and Non-Euclidean geometries

Special Relativity

Introduction to topology

5.1. Galois connections

5.2. Monotone Galois connections (adjunctions)

5.3. Upper and lower bounds

5.4. Complete lattices

5.5. Fixed point theorem

5.6. Transport of closure

5.7. Preorder generated by a relation

5.8. Finite sets

5.9. Generated equivalence relations, and more

5.10. Well-founded relations

Complements:5.2. Monotone Galois connections (adjunctions)

5.3. Upper and lower bounds

5.4. Complete lattices

5.5. Fixed point theorem

5.6. Transport of closure

5.7. Preorder generated by a relation

5.8. Finite sets

5.9. Generated equivalence relations, and more

5.10. Well-founded relations

Cardinals6. Universal algebra

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

Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system (Requires to have read the above metamathematical complements)

(to be completed)

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.

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

Research teams and centers : Europe - North America - Other

Publications
- Blogs - Organizations
- Mailing
lists - Software
- Other

Criticism of the academic system

Criticism of the academic system

List of
physics theories that will progressively link to other
pages presenting each theory. The main presentations of
physics theories already available here are

- Quantum physics
- (A presentation of Special
Relativity Theory is still in draft)

- Tensors.
- General relativity:

An
exploration of physics by dimensional analysis : telling a
lot of fundamental physics and the amplitudes of diverse effects
mainly by multiplying quantities. Some sections have been
moved to separate pages:

The speed of light and astronomical distancesSolved physics problems (for now just one thing about gravitation)

The energy of nuclear reactions ; The radius of nuclei

Electromagnetism

The gravitational constant

Effects of General Relativity

The parameters of the atomic structure

The compressibility of condensed matter

The speed of the sound in condensed matter

Temperature(to be continued)