Set theory and Foundations of Mathematics |
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| 1. First foundations of mathematics | |
| 2. Set theory (all in one file - pdf) | |
| 2.1. First axioms of set theory | The inclusion predicate Formulas vs statements The role of axioms Converting the binders Classification of axioms Axioms for notions Axiom of Extensionality Axioms for functions |
| 2.2. Set generation principle | Formalizing diverse notions in set theory Sets as domains of bounded quantifiers Statement of the principle Main examples (union, image, ∅, pairs...) |
| 2.3. Currying and tuples
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Formalizing operations and relations Currying Tuples Tuples axiom Tuples in set theory |
| 2.4. Uniqueness quantifiers
and conditional operators |
Uniqueness quantifiers Single element axiom Conditional connective Conditional operator Relations as operations Defining the tuples definers |
| 2.5. Families, Boolean operators on sets | Families Structures and binders Extensional definition of sets Union of a family of sets Other Boolean operators on sets Intersection |
| 2.6. Graphs | Currying notation Functional graphs Indexed partitions Sum or disjoint union Direct and inverse images by a graph Direct and inverse images by a function |
| 2.7. Products and powerset | Cartesian product of two sets Finite products, operations and relations Translating operators into predicates More primitive symbols (Powerset, Exponentiation, Product) Their equivalence Cantor's Theorem The ZF approach |
| 2.8. Injectivity and inversion | Composition, restriction Injections, bijections, inverse Diverse properties Canonical functions Sum of functions Product of functions or recurrying |
| 2.9. Binary relations on a set | Preimages and products Basic properties Preorders and orders Strict and total orders Equivalence relations Quotient functions Partitions |
| 2.10. Axiom of choice | Properties of curried composition Axiom of choice over a set (ACX) Dependencies between diverse ACX More statements simply equivalent to AC |
| 2.A. Time in set theory | Standard universes The standardness ideal The realistic view of set theory Standard multiverses Can a set contain itself ? |
| 2.B. Interpretation of classes | The relative sense of open quantifiers
The indefiniteness of classes Classes and sets in expanding universes Justifying the set generation principle Concrete examples |
| 2.C. Concepts of truth in mathematics | Arithmetic truths More concepts of strength Set theory from realism to axiomatization Axioms compatibility condition Alternative logical frameworks |
| 3. Algebra - 4. Arithmetic - 5. Second-order foundations | |