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. 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 (12 + 6 pages - full text in 1 html page).
1.1. Introduction to the foundation of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories: notions, objects and meta-objects
1.4. Structures of mathematical systems
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 (last update : april 2015, only in this html page ; pdf not updated yet)
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. Mathematical theories and the Completeness Theorem
3.2. How mathematical theories develop
3.3. Morphisms of relational systems and concrete categories
3.4. Algebras
3.5. Products of systems
3.6. Integers and recursion (new, May 2015)
3.7. Second-order theories (updated, May 2015)
3.8. Formalizations of Arithmetic
3.9. Non-standard models of Arithmetic
3.10. The Incompleteness Theorem

More philosophical notes : About the powerset axiom

4. Algebra and geometry
(List of texts on algebra)
The Galois connection between structures and permutations (Automorphisms, Invariants)
Introduction to categories
Monoids and groups
Actions of monoids and groups
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

General formalization tools (draft)
Polymorphisms and invariants
Relational clones
(To be continued - see below drafts)

5. Galois connections (11 pdf pages). Makes rigorously no use of parts 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). The beginning was moved to 2.11.

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

Drafts of more texts, to be reworked later

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

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

Diverse texts ready but not classified

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

Resolution of cubic equations

Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system (Requires to have read Part 1 with philosophical aspects)

Contributions to Wikipedia

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.

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
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

(to be continued)

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

Other topics

Why I am upset : I cannot find anyone both able to think and interested to make sense of life ; either way, people prefer to die stupid than to make a better world.
On the world's failures to steer itself properly : a very long analysis of the stakes of the future and how they are often improperly considered in essays of 2014's contest "How should humanity steer the future", with a few sections in separate pages: Links against psychiatry