Set Theory and Foundations of Mathematics

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Other languages : FRRUES


        of foundations
1. First foundations of mathematics (detailed list of sections), with updates until Nov. 2015 (while the pdf version in 13 + 7 pages, and full text in 1 html page, were not recently updated).
1.1. Introduction to the foundation 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
Time in set theory
Interpretation of classes
Concepts of truth in mathematics

2. Set theory (continued) (updated html version, April 2016; 11 pdf pages not fully updated)
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 on a set ; ordered sets
2.8. Canonical bijections
2.9. Equivalence relations and partitions
2.10. Axiom of choice
2.11. Notion of Galois connection

3. Model Theory (3.1 to 3.5 updated, March 2016)
3.1. Morphisms of relational systems and concrete categories
3.2. Special morphisms
3.3. Algebras
3.4. Algebraic terms and term algebras
3.5. Integers and recursion
3.6. Arithmetic with addition (updated, April 2016)
3.7. The Completeness Theorem
3.8. How theories develop
3.9. Second-order logic
3.10. Second-order arithmetic
3.11. Non-standard models of Arithmetic
3.12. The Incompleteness Theorem
More philosophical notes :
Philosophical proof of consistency of the Zermelo-Fraenkel axiomatic system (uses Part 1 with philosophical aspects + recursion)

4. Algebra and geometry
(List of texts on algebra)
The Galois connection between structures and permutations (Automorphisms, Invariants)
Introduction to categories
Monoids and groups
Actions of monoids and groups
Introduction to the foundations of geometry
What is geometry
Structures and permutations in the plane
Affine geometry
Beyond affine geometry
Euclidean geometry
The completeness of first-order geometry

General formalization tools (draft)
Products of systems
Polymorphisms, invariants and clones of operations
Relational clones
Abstract clones
(To be continued - see below drafts)

5. Galois connections (11 pdf pages). Makes rigorously no use of parts 3 and 4, but only uses text 1 (without complements) and 2. Its position has been moved from 3 to 5 for pedagogical reasons (higher difficulty level while the above texts 3 and 4 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
Duality systems and theories
Affine geometry
Introduction to topology
Vector spaces in duality
Axiomatic expressions of Euclidean and Non-Euclidean geometries

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

Contributions to Wikipedia

I wrote large parts of the Wikipedia article on Foundations of mathematics (September 2012 - because until then, 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.