Set Theory and Foundations of Mathematics

.

About this site (purpose and author)
Presentation videos : Why learn Physics by yourself.
This site in other languages : FrenchRussian Cycle
        of foundations
About html display of symbols / wrong links
(I need help for several tasks : some only require to be a good internet user ; for others I need translators or web programmers. Please write me if you can help either for free or as a small job - but do not expect US or west european salaries, thanks)

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
Philosophical aspects
Time in model theory
Time in set theory

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


3. Model Theory

3.1. What is a mathematical theory
3.2. How mathematical theories develop
3.3. Relational systems and morphisms
3.4. Algebras
3.5. The Galois connection between (invariant) structures and permutations (automorphisms) (updated august 2014)
3.6. Second-order theories
3.7. Formalizations of Arithmetic
3.8. Non-standard models of Arithmetic
3.9. The Incompleteness Theorem
More philosophical notes : About the powerset axiom

4. Algebra and geometry

(List of texts on algebra)

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
Products of relational systems
Truth of formulas in products
Morphisms into products
Products of algebras

Polymorphisms and invariants
The Galois connection Inv-Pol between sets of operations and relations
The Galois connection Pol-Pol between sets of operations

Duality theories (to be completed)
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 monoids
Monoids, theories and morphisms
Groups

Vector spaces in duality

Dimensional analysis
Axiomatic expressions of Euclidean and Non-Euclidean geometries

Special Relativity

Introduction to topology

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).

5. Galois connections
(incomplete for now : 6 pdf pages up to 5.6. ; should have about 11 pages when done). Due to a partial rework, the 5.1 and 5.2 partially repeat 2.11.
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:
Cardinals

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)
6. Universal algebra
(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))

Resolution of cubic equations


Directory of links : Logic and set theory around the world

Research teams and centers : Europe - North America - Other

Publications - Blogs - Organizations - Mailing lists - Software - Other
Criticism of the academic system


Foundations of theoretical physics

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

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 distances
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)

Solved physics problems (for now just one thing about gravitation)