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 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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send feedback and suggestions to bombelli at olemiss.edu – modified 9 feb 2021