A new approach to set theory

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...