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*
(*A* – *B* is not a good notation, in general (*A* \ *B*) ∪ *B* ≠ *A*).

* __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__:
*a*^{n}* a*^{m}
= *a*^{n+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 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 *χ*_{A}: *X* → [0,1],
which is interpreted, for each *x* ∈ *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]; > s.a. clifford algebra; probabilities in physics.

@ __Non-Cantorian set theory__: Cohen & Hersh SA(67)dec.

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send feedback and suggestions to bombelli at olemiss.edu – modified 10
may
2017