In
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 concer ...
a serial relation is a
homogeneous relation expressing the connection of an element of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
to the following element. The
successor function used by
Peano to define
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s is the prototype for a serial relation.
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
used serial relations in ''
The Principles of Mathematics''
(1903) as he explored the foundations of
order theory and its applications. The term ''serial relation'' was also used by
B. A. Bernstein
Benjamin Abram Bernstein (20 May 1881, Pasvalys, Lithuania – 25 September 1964, Berkeley, California) was an American mathematician, specializing in mathematical logic.
Biography
With his Jewish family, Bernstein immigrated as a child to the Uni ...
for an article showing that particular common axioms in order theory are nearly incompatible:
connectedness, irreflexivity, and
transitivity.
A serial relation R is an
endorelation on a set ''U''. As stated by Russell,
where the universal and existential quantifiers refer to ''U''. In contemporary language of
relations, this property defines a
total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.
For a relation ''R'', let denote the "successor neighborhood" of ''x''. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".
In
normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.
Russell's series
Relations are used to develop series in ''The Principles of Mathematics''. The prototype is Peano's
successor function as a one-one relation on the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. Russell's series may be finite or generated by a relation giving
cyclic order. In that case, the
point-pair separation relation is used for description. To define a progression, he requires the generating relation to be a
connected relation. Then
ordinal numbers are derived from progressions, the finite ones are finite ordinals.
Distinguishing open and closed series
results in four total orders: finite, one end, no end and open, and no end and closed.
Contrary to other writers, Russell admits negative ordinals. For motivation, consider the scales of
measurement using
scientific notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
, where a power of ten represents a
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "d ...
of measure. Informally, this parameter corresponds to
orders of magnitude used to quantify physical units. The parameter takes on negative as well as positive values.
Stretch
Russell adopted the term ''stretch'' from
Meinong, who had contributed to the theory of distance. Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."
To explain Meinong, Russell refers to the
Cayley-Klein metric, which uses stretch coordinates in
anharmonic ratios which determine distance by using logarithm.
[Russell (1897) ''An Essay on the Foundations of Geometry'']
References
External links
* Here: page 416.
* {{cite journal, last = Yao, first = Y.Y., author2=Wong, S.K.M., title = Generalization of rough sets using relationships between attribute values, journal = Proceedings of the 2nd Annual Joint Conference on Information Sciences, year = 1995, pages = 30–33, url = http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf.
Binary relations
Order theory