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Set theory and Foundations of Mathematics
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	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

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