Set theory and Foundations of Mathematics 

1.
First
foundations of mathematics 

2.
Set theory (continued) (old pdf version : 11 pages) 

2.1. Formalization of set theory  The role of axioms
Converting binders Open quantifiers, formulas vs statements The inclusion predicate Axioms for notions Axiom of Extensionality Axioms for functions Formalizing diverse notions in set theory 
2.2. Set generation principle  The role of sets Expression of the principle List of cases: union, image, ∅, pairs... 
2.3. Tuples, families

Formalization of operations and currying Tuples Conditional connective Families Structures and binders Extensional definition of sets 
2.4. Boolean operators on families of sets  Union of a family of sets Intersection Other operators 
2.5. Products, graphs and composition  Finite product Sum or disjoint union Composition, restriction, graph of a function Direct image, inverse image 
2.6. Uniqueness quantifiers, functional graphs  Definitions of the uniqueness quantifiers Translating operators into predicates Conditional operator Functional graphs 
2.7. The powerset axiom  Need of additional primitive symbols Powerset Exponentiation Product Cantor Theorem 
2.8. Injectivity and inversion  Injections Inverse Properties of injectivity, surjectivity, composition and inversion 
2.9. Properties
of binary relations on a set ; ordered sets 
Properties of binary relations Preorder Order Strict order, total order Monotone and antitone functions Order on sets of functions 
2.10. Canonical bijections  Notion of canonical bijection Sum of sets, sum of functions Product of functions or recurrying 
2.11. Equivalence relations and partitions  Indexed partitions Equivalence relation defined by a function Partition, canonical surjection Order quotient of a preorder 
2.12. Axiom of choice  6 simply equivalent formulations 
2.13. Notion
of Galois connection (basic properties ; see text Galois connections for more developments) 
Fixed points; idempotence Notion of Galois connection General example Properties Closures 
Time in set theory  The expansion of the set
theoretical universe Can a set contain itself ? The relative sense of open quantifiers 
Interpretation of classes  Classes in an expanding universe Concrete examples Justifying the set generation principle 
Concepts of truth in mathematics  Provability Arithmetic truths Set theoretical truths Alternative logical frameworks 