Set theory and Foundations of Mathematics

1. First foundations of mathematics

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Previous versions in PDF (13 pages for main text + 7 pages for philosophical aspects).
1.1. Introduction to the foundations of mathematics
Mathematics and theories
Foundations and developments
The cycle of foundations
1.2. Variables, sets, functions and operations
Free and bound variables
Ranges and sets
Philosophical aspects
(Each subsection assumes those on its left and top)

Intuitive representation and abstraction
Platonism vs Formalism
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
Set-theoretical interpretation
Realistic vs. axiomatic theories in
mathematics and other sciences
1.4. Structures of mathematical systems
Structures of set theory
About of ZF set theory
Types in one-model theory
The notion of structure in one-model theory

1.5. Expressions and definable structures
Terms and formulas
Variable structures
Structures defined by expressions
Invariant structures
Time in model theory
The time of interpretation
The metaphor of the usual time
The finite time between expressions
1.6. Logical connectives
Conjunctions, disjunctions
Chains of implications and equivalences
Axioms of equality
1.7. Classes in set theory
The unified framework of theories
Classes, sets and meta-sets
The infinite time between theories
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
Status of open quantifiers in set theory
The relative sense of open quantifiers
Interpretation of classes
Classes in an expanding universe
Concrete examples
1.10. Formalization of set theory
The inclusion predicate
Translating the definer into first-order logic
First axioms
A general principle for the formalization of set theory
Formalization of operations and currying
1.11. Set generation principle
Criterion to accept some classes as sets :
Subsets, union, image, �, pairs
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 : Th�orie des ensembles et fondements des math�matiques : Premiers fondements