Set Theory and Foundations of Mathematics

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Other languages : FRRUTRES


Cycle
        of foundations
1. First foundations of mathematics (details) - all in 1 file (35 paper pages) - obsolete pdf in 13 + 7 pages.
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories: notions, objects, meta-objects
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
1.6. Logical connectives
1.7. Classes in set theory
1.8. Bound variables in set theory
1.9. Quantifiers
1.10. Formalization of set theory
1.11. Set generation principle
Philosophical aspects
Time in model theory
Truth undefinability
Time in set theory
Interpretation of classes
Concepts of truth in mathematics

2. Set theory (continued) - all in one file (17 paper pages; obsolete pdf in 11 pages)
2.1. Tuples, families
2.2. Boolean operators on families of sets
2.3. Products, graphs and composition
2.4. Uniqueness quantifiers, functional graphs
2.5. The powerset axiom
2.6. Injectivity and inversion
2.7. Properties of binary relations ; ordered sets
2.8. Canonical bijections
2.9. Equivalence relations and partitions
2.10. Axiom of choice
2.11. Galois connection

The following parts are undergoing some restructuring.
3. Algebra 1 (all in one file)
3.1. Relational systems and concrete categories
3.2. Algebras
3.3. Special morphisms
3.4. Monoids
3.5. Actions of monoids
3.6. Invertibility and groups
3.7. Categories
3.8. Initial and final objects
3.9. Eggs, basis, clones and varieties

4. Arithmetic and first-order foundations (all in one file)
4.1. Algebraic terms
4.2. Term algebras (still incomplete)
4.3. Integers and recursion
4.4. Presburger Arithmetic
4.5. Finiteness and countability (draft)
4.6. The Completeness Theorem
4.7. Non-standard models of Arithmetic
4.8. Developing theories : definitions
4.9. Constructions

5. Second-order foundations
5.1. Second-order invariants
5.2. Second-order logic
5.3. Well-foundedness
5.4. Ordinals and cardinals (draft)
5.5. Undecidability of the axiom of choice
5.6. Second-order arithmetic
5.7. The Incompleteness Theorem (draft)
More philosophical notes (uses Part 1 with philosophical aspects + recursion) :
Gödelian arguments against mechanism : what was wrong and how to do instead
Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system

6. Foundations of Geometry (draft)
6.1. Introduction to the foundations of geometry
6.2. First-order invariants in concrete categories
6.3. Affine spaces
5.5. Duality
5.6. Vector spaces and barycenters
Beyond affine geometry
Euclidean geometry

7. Algebra 2 (draft)
Products of systems
Varieties
Polymorphisms and invariants
Relational clones
Abstract clones
Rings
(To be continued - see below drafts)

Galois connections (11 pdf pages). Rigorously it only uses parts 1 (without complements) and 2. Its position has been moved from 3 for pedagogical reasons (higher difficulty level while the later texts are more directly interesting). The beginning was moved to 2.11.

Monotone Galois connections (adjunctions)
Upper and lower bounds, infimum and supremum
Complete lattices
Fixed point theorem
Transport of closure
Preorder generated by a relation
Finite sets
Generated equivalence relations, and more
Well-founded relations

Drafts of more texts, to be reworked later

Dimensional analysis : Quantities and real numbers - incomplete draft text of a video lecture I wish to make on 1-dimensional geometry
Introduction to inversive geometry
Affine geometry
Introduction to topology
Axiomatic expressions of Euclidean and Non-Euclidean geometries
Cardinals

Well-orderings and ordinals (with an alternative to Zorn's Lemma).

Diverse texts ready but not classified

Pythagorean triples (triples of integers (a,b,c) forming the sides of a right triangle, such as (3,4,5))
Resolution of cubic equations
Outer automorphisms of S6

Contributions to Wikipedia

I wrote large parts of the Wikipedia article on Foundations of mathematics (Sep. 2012 - before that, 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.