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 classesClasses 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
3. Model theory
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