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, others are philosophical
1.1. Introduction to the foundations of mathematics What is mathematics
Foundations and developments
Platonism vs Formalism
The cycle of foundations
1.2. Variables, sets, functions and operations Constants
Free and bound variables
Ranges and sets
1.3. Form of theories The variability of the model
Notions and objects
One-model theory
The diversity of logical frameworks
Examples of notions from various theories
Components of theories
Set-theoretical interpretation
1.4. Structures of mathematical systems 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
Display conventions
Variable structures
Structures defined by expressions
Invariant structures
The role of axioms
1.6. Logical connectives
Conjunctions, disjunctions
Chains of implications and equivalences
1.7. Classes in set theory Classes, sets and meta-sets
Definiteness classes
Extended definiteness
1.8. Binders in set theory The syntax of binders
Russell's Paradox
The function definer
1.9. Axioms and proofs
Realistic vs. axiomatic theories in mathematics and other sciences
Completeness of first-order logic
1.10. Quantifiers The quantifiers ∃,∀
Inclusion between classes
Rules of proofs for quantifiers
Second-order quantification
Axioms of equality
Time in model theory The time of interpretation
The metaphor of the usual time
The infinite time between models
Truth undefinability
The Berry paradox
Set theory as a unified framework Structure definers in diverse theories
The unified framework of theories
Set theory as a unified framework of itself
Zeno's Paradox
2. Set theory
3. Algebra
4. Arithmetic
5. Second-order foundations

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