Set theory and Foundations of Mathematics

1. First foundations of mathematics

The last updated version is as the below list of separate html pages and also all in 1 html page.
Previous versions in PDF (13 pages for main text + 7 pages for philosophical aspects).
Some pages focus on model theory, others focus on set theory.
1.1. Introduction to the foundations of mathematics
Mathematics and theories
Foundations and developments
The cycle of foundations
Philosophical aspects
(Each subsection assumes those on its left and top)

Intuitive representation and abstraction
Platonism vs Formalism
1.2. Variables, sets, functions and operations
Free and bound variables
Ranges and sets

1.3. Form of theories
The variability of the model
Notions and objects
One-model theory
On the diversity of formalisms
Examples of notions from various theories
Components of theories
Realistic vs. axiomatic theories in
mathematics and other sciences
1.4. Structures of mathematical systems
Set-theoretical interpretation
First-order structures
Structures of set theory
About of ZF set theory

Formalizing types and structures as objects of one-model theory

1.5. Expressions and definable structures
Terms and formulas
Variable structures
Structures defined by expressions
Invariant structures
The role of axioms
Time in model theory
The time of interpretation
The metaphor of the usual time
The infinite time between models
1.6. Logical connectives
Conjunctions, disjunctions
Chains of implications and equivalences
1.7. Classes in set theory
The unified framework of theories
Classes, sets and meta-sets
Truth undefinability

The Berry paradox
Zeno's Paradox
Definiteness classes
Extended definiteness
1.8. Binders in set theory
The syntax of binders
Definition of functions by terms
Relations and set-builder symbol
Russell's Paradox
Time in set theory
The expansion of the set theoretical universe
Can a set contain itself ?
1.9. Quantifiers
The quantifiers ∃,∀
Inclusion between classes
Rules of proofs for quantifiers
Second-order quantification
Axioms of equality
The relative sense of open quantifiers
Interpretation of classes
Classes in an expanding universe
Concrete examples
1.10. Formalization of set theory
The role of axioms
Open quantifiers, formulas vs statements
Converting binders
Axioms for notions
The inclusion predicate
Axioms for functions
1.11. Set generation principle
Criterion to accept some classes as sets :
Subsets, union, image, ⌀, pairs
A general principle for the formalization of set theory
Formalization of operations and currying
Justifying the set generation principle

Concepts of truth in mathematics
Arithmetic truths
Set theoretical truths
Alternative logical frameworks
2. Set theory (continued)
3. Model theory

Other languages :
FR : Théorie des ensembles et fondement des mathématiques : Premiers fondements