Set theory
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 obedience to the Zermelo-Fraenkel theory or other traditional systems of axioms, 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 70 pages ready, 30 more pages in draft, and 50 to 100 more pages in plans). The translation into English is just starting.
The ambition of this approach is to be as much as possible simultaneously deep, elegant and rigourous, much more than any other existing approach. Only the second part will be very close to 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.
If anyone would like to help translating contents from French, it would be very helpful to speed up the development of this project. (Contact : trustforum at gmail.com).
Here will be the contents of the first 60 pages:
1. Naive set theory : 23 pdf pages, split into 4 html files.
1.1.
What is mathematical logic
1.2. About set theory
1.3. Variables
and sets
1.4. Maps, unary relations and comprehension
1.5. The paradox of language and its time explanation
1.6. Russell Paradox and explanation (notion of class)
1.7. Finite sets, tuples, families, product
1.8. Operations, relations
1.9. Construction of terms and sentences
2. First developments (18 pdf pages)
2.1. Some properties of quantifiers
2.2. Operations on sets; the powerset axiom
2.3. Study of maps
2.4. Canonical bijections
2.5. Notions on binary relations
2.6. Study of equivalence relations
2.7. Axiom of choice
3. Galois connections (18 pdf pages)
3.1. Notions on ordered sets, Galois connections
3.2. Monotone Galois connections
3.3. Upper and lower bounds
3.4. Complete lattices
3.5. Fixed point theorem
3.6. Preorder generated by a relation
3.7. Finite sets
3.8. Generated equivalence relations, and others
3.9. Well-founded relations