(The English vocabulary used here may need little corrections as
I'm not native English speaker and I need to name some concepts
that have not always a standard name in the mathematical
literature; I know that my use of "functor" differs from (is more
general than) its standard meaning in category theory, and thus
breaks the standard convention, but I do not know any better word
for what I need).
A first version was written in French,
then partially corrected and translated to make the start of this
site, which then continued directly in English on other subjects.
I look
for translators to provide versions of this work
in other languages.
Now over 50 pages ready from the start on mathematics (entirely self-contained), plus more sections of maths and physics in bulk or draft; still incomplete (some French pages should be corrected before translation, and others are in plans to be directly written here).
The set theory formalism presented here differs from the
traditional ZF system, but aims to better fit the common use of
mathematics and be cleaner on foundation and meaning ; after
developing this work I noticed that this kind of approach seems to
have been already explored and presented by other author(s), under
the name "functional set theory" but I did not study their works.
This work is generally intended to be slowly read,
respecting the order
1. Beginnings of set theory (html) - pdf version (10 pages - updated on 20/5).
The flow of
the metamathematical time (html)
(Philosophical complements, not needed for text 2 but only to
intuitively justify the above generation principle, and for the
comments on the powerset in text 3)
Note on alternative logics
2. More notions of set theory (10 pdf pages)
What is a mathematical theoryMore philosophical notes : About the powerset axiom, including
The contents of a theoryNotion of Algebra
Provability and the Completeness Theorem
How mathematical theories develop
Proofs
Definitions
Constructions
Morphisms, subalgebras, images...Relational systems and their morphisms
Transformation monoids
Permutation groups
Commutants
EmbeddingsThe Galois connection between (invariant) structures and permutations (automorphisms)
Algebraic formulas and theories
More similarities between development levelsSecond-order theories
Second-order and higher-order theoriesFormalizations of Arithmetic
Translating second-order theories into first-order ones
Second-order structures
The expression in set theoryNon-standard models of Arithmetic
Second-order arithmetic
First-order arithmetic
Non-standard models of elementary arithmeticThe Incompleteness Theorem
Non-standard models of Presburger Arithmetic
Non-standard models of full first-order arithmetic
On the double meaning of invariance
The Truth Undefinability Theorem
The Incompleteness Theorem
The need of the powerset axiom
Its fundamentally incomplete meaning (Skolem's paradox)
About the axiom of choice
(To be completed: Strength hierarchy of set theories).
4. Algebra and geometry
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
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)
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
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 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)
Contact : trustforum at gmail.com