Set Theory  

In General
* Idea: A branch of mathematical logic that studies sets, which informally are collections of objects, independently of the nature of their constituents.
* History: Often distinguished into combinatorial set theory (grew around Erdős et al in Budapest) and axiomatic set theory (uses more mathematical logic, topology and measure theory; Developed with Cohen, Scott & Solovay at Stanford and Berkeley); The two are discovering each other roughly after the work of Silver.
* Types of structures: Binary operation; Relation; Topology.
@ General references: Kamke 50; Bourbaki L1; Cohen 66; Jech 71; Halmos 74; Williams 77; Devlin 79; Kunen 80; Shelah BAMS(03) [status]; Schimmerling 11; in Vermani & Vermani 12 [discrete mathematics]; Dasgupta 14 [IIb]; Cunningham 16 [text, IIb].
@ Special emphasis: Rudeanu 12 [and order structures].
@ And physics: Svozil FP(95); Titani & Kozawa IJTP(03).
@ Philosophical: Tiles 89; Muller BJPS(01) [classes and categories]; > s.a. Infinite.
> Online resources: see Stanford Encyclopedia of Philosophy page; Wikipedia page.

Special Sets and Types of Sets
* Empty set: The symbol Ø was introduced in 1937 by André Weil, the only member of the Bourbaki group who knew the Norwegian alphabet.
* Directed set: A set X with a reflexive, transitive relation ≤ such that for all x and x' in X, there exists exists an x'' in X with x, x' ≤ x''; > s.a. Net.
@ Directed set: in Eilenberg & Steenrod 52.
@ Other types: Gouéré mp/02, CMP(05) [almost periodic discrete sets].

Special Subsets > see Filter.

Operations on Sets > s.a. algebra [Jordan algebras]; Ternary Operations.
* Elementary operations: Intersection, union, Cartesian product; Set difference A \ B (AB is not a good notation, in general (A \ B) ∪ BA).
* One-point union: The one-point union of X and Y is X V Y:= X × {x} ∪x=y {y} × Y; Looks like X and Y joined at a point x = y.
$ Associative operation: A binary operation (a, b) → a \(\circ\) b is associative if \((a \circ b) \circ c = a \circ (b \circ c) = a \circ b \circ c\) for all a, b, c.
* Generalization of associative operation: an am = an+m; All Jordan algebras satisfy this condition.
* Example of non-associative binary operation: Cross product of vectors in \(\mathbb R\)3; > s.a. Non-Associative Geometry.
@ References: Sanders BAMS(13) [structure theory of set addition].

Related Concepts and Techniques > see Russell Paradox; Venn Diagrams.

Fuzzy Set Theory
* Idea: A generalization of set theory, founded in 1965 by Lotfi Asker Zadeh; Its motivation is to include the notion of uncertainty about whether some object satisfies some property.
$ Fuzzy subset: Given a set X, a fuzzy subset \(A\) of \(X\) is a function \(\chi^{~}_A: X \to [0,1]\), which is interpreted, for each \(x \in M\), as the probability that x is included in A.
@ References: Kandel 86; Novak 87; Lowen 96; García-Morales a1704 [new approach]; > s.a. logic; set of posets [fuzzy ordered sets].
> Online resources: see Wikipedia page.

Other Generalizations > s.a. Quasi-Set Theory; Topos Theory.
* Examples: Some other examples are quasi-set theory, and evolving sets.
@ Quantum sets: Ozawa JSL(07)-math/06; Benavides a1111 [and sheaf logic]; Finkelstein a1403, a1403 [quantum set algebra]; Ozawa NGC(16)-a1504 [and the probabilistic interpretation of quantum theory]; Kornell a1804 [in non-commutative geometry]; Azawa APAL-a2002; Ozawa a2102 [work of Gaisi Takeuti]; > s.a. clifford algebra; probabilities in physics.
@ Non-Cantorian set theory: Cohen & Hersh SA(67)dec.

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