The foundations of mathematics
by Sylvain Poirier
This site under construction will be dedicated to presenting a new
vision of set theory and the foundations of mathematics, to provide a
deeper understanding of the subject beyond the usual raw acceptance of
the Zermelo-Fraenkel axioms system (or other traditional axioms
systems), and to conveniently present insights to the foundations of
algebra, based on concepts of universal algebra that will be presented
in a simpler approach than the usual universal algebra textbooks.
This new approach started to be written in
French
and is still incomplete there (about 50 pages nearly ready, 30 more
pages in
draft, and 50 to 100 more pages in plans). The translation into English
of what is ready, just started and will be progressively completed in
the next few weeks.
The ambition of this approach is to combine the following advantages by
a new structure of progression:
- It is perfectly rigorous, all what is provable is proven
from
the beginning. (Bourbaki had this goal, but here further goals are
added, which might seem a priori conflicting with this one; better than
a compromise, I offer here a new intergrated solution to gather several
more advantages)
- Conduct original approaches to most subjects: most concepts
were
already largely known here, there, but many of them had not, as far i
know, been described and articulated in this way. Except about a half
the contents of text 2, which has much in common with traditional
courses, other parts also have some common aspects with what can be
found elsewhere, but it comes with new structures and explanations,
simplified proofs, and the like.
- Emphasis is placed on intuition, the "philosophical"
explanation, the deeper meaning of things, the mathematical world,
without being slowed down by the development of tedious formal rules.
Here, rigour or formalisation will neither be missing nor dilute or
obscure meaning. On the contrary, the right formalism will express the
highest concentration of meaning.
- The tools developed are very powerful and general
- Very few proofs take more than half a page, and most proofs
are
just couple of lines long; all choices are made with great care for the
shortest and elegant path as possible, better than any other existing
approach.
If anyone would like to help translating contents from French,
it
would be very helpful to speed up the development of this project. (Author
contact : trustforum at gmail.com).

Here will be the contents of the first 50 pages (already written in
French):
1. Set theory (start) (html - pdf) -
(translation from French in progress - the French version in pdf has
currently 11 + 6 pages).
1.1. Introduction to the foundation of
mathematics
1.2. Variables, sets, functions and operations
1.3. Structure of theories: objects, meta-objects, types and structures
1.4. Terms and formulas; connectives
1.5. Classes in set theory
1.6. Bound variables in set theory
1.7. Quantifiers
1.8. First set theory axioms
1.9. Set generation principle
Metamathematical Complements
1.A. Completeness and incompleteness theorems
1.B. The time metaphor of foundations
1.C. The Zenon paradox as a metaphor of set theory
1.D. Interpretation of classes
1.E. The meaning of open quantifiers
1.F. Justifying the set generation principle
1.G. Can a set be an element of itself ?
1.H. Remark on nonstandard logics
1.I. Concrete examples
2. First developments (17 pdf pages)
2.1. More quantifiers
2.2. Tuples, families;
2.3. Operators on sets
2.4. The powerset axiom
2.5. Injections, surjections, canonical bijections
2.6. Other properties of maps
2.7. Binary relations on a set
2.8. Study of equivalence relations
2.9. Notions on ordered sets
2.10. Axiom of choice
3. Galois connections (16 pdf pages)
3.1. Galois
connections
3.2. Monotone Galois connections
3.3. Upper and lower bounds
3.4. Complete lattices
3.5. Fixed point theorem
3.6. Transport of closure
3.7. Preorder generated by a relation
3.8. Finite sets
3.9. Generated equivalence relations, and more
3.10. Well-founded relations
4. Universal algebra I
Of course, this list of first sections titles are not enough to express
what are the interest and innovation here. You'd need to see the full
text to figure out, which for now requires you to know French...