Set Theory| 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:
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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 9 feb 2021