set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a semiset is a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
.
The theory of semisets was proposed and developed by
Czech
Czech may refer to:
* Anything from or related to the Czech Republic, a country in Europe
** Czech language
** Czechs, the people of the area
** Czech culture
** Czech cuisine
* One of three mythical brothers, Lech, Czech, and Rus'
Places
* Czech, ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s
Petr Vopěnka
Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of articles and m ...
and
Petr Hájek
Petr Hájek (; 6 February 1940 – 26 December 2016) was a Czech scientist in the area of mathematical logic and a professor of mathematics. Born in Prague, he worked at the Institute of Computer Science at the Academy of Sciences of the Czech Repu ...
(1972). It is based on a modification of the
von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collect ...
; in standard NBG, the existence of semisets is precluded by the
axiom of separation
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
.
The concept of semisets opens the way for a formulation of an
alternative set theory
In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set ...
.
In particular, Vopěnka's
Alternative Set Theory
In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set ...
(1979) axiomatizes the concept of semiset, supplemented with several additional principles.
Semisets can be used to represent sets with imprecise boundaries. Novák (1984) studied approximation of semisets by
fuzzy set
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined a ...
s, which are often more suitable for practical applications of the modeling of imprecision.
References
*Vopěnka, P., and Hájek, P. ''The Theory of Semisets''. Amsterdam: North-Holland, 1972.
*Vopěnka, P. ''Mathematics in the Alternative Set Theory.'' Teubner, Leipzig, 1979.
*Holmes, M.R. Alternative Axiomatic Set Theories, §9.2 Vopenka's alternative set theory. In E. N. Zalta (ed.): ''
The Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
'' (Fall 2014 Edition).
*Novák, V. "Fuzzy sets—the approximation of semisets." ''
Fuzzy Sets and Systems
''Fuzzy Sets and Systems'' is a peer-reviewed international scientific journal published by Elsevier on behalf of the International Fuzzy Systems Association (IFSA) and was founded in 1978. The editors-in-chief (as of 2010) are Bernard De Baets ...