Set Theory and the Foundations of Mathematics
by Sylvain Poirier
This work aims to rebuild mathematics in deep, clarified and optimized
ways, from model theory and a new formalization of set theory, to
universal algebra and tensors.
This new approach started to be written in
French
and is still incomplete there (about 40 pages nearly ready, 20 or 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
could help to speed up the development of this project. Also, the English vocabulary used here may need little corrections as I'm not native English speaker and I need to name some concepts that may sometimes be not very common. Example: if E is the
set of possible values of a variable x, do we say E is the "domain" or
the "range" of x (or another name) ?)
Overview of some features of the foundations of mathematics, focusing on the philosophical aspects, with metaphysical consequences. (shorter, more general and less rigorous than the below)
Here will be the contents of the first 40 pages or so (already written
in
French):
1. Set theory (start) (html - pdf) -
(14 pages: 10 for the main sections + 4 for complements).
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. Metamathematical time and Zeno's Paradox
1.C. The relative sense of open quantifiers
1.D. Nature of classes and the set generation principle
1.E. Concrete examples
1.F. Can a set be an element of itself ?
1.G. Note on alternative logics
(Another section "Force hierarchy of set theories" instead of being included here will be later integrated in the above overview).
2. Elementary constructions (translation in progress -
ultimately 12 pdf pages ?)
2.1. Tuples, families
2.2. Operators on sets
2.3. Uniqueness quantifier
2.4. Properties of functions
2.5. The powerset axiom
2.6. Canonical bijections
2.7. Binary relations on a set
2.8. Equivalence relations and partitions
2.9. Axiom of choice
2.10. Notions on ordered sets
3. Galois connections (12 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
Here is a list of physical theories that will progressively link to other pages presenting each theory. These already include an introduction to quantum physics.
Contact : trustforum
at gmail.com