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 concern ...

a serial relation is a

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) 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 In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...

used by

Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...

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, a ...

used serial relations in '' The Principles of Mathematics'' (1903) as he explored the foundations of

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

and its applications. The term ''serial relation'' was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible:

connectedness In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...

, irreflexivity, and transitivity. A serial relation R is an

endorelation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

on a set ''U''. As stated by Russell,

\forall x \exists  y \ xRy ,

where the universal and existential quantifiers refer to ''U''. In contemporary language of relations, this property defines a

total relation In mathematics, a binary relation ''R'' ⊆ ''X''×''Y'' between two sets ''X'' and ''Y'' is total (or left total) if the source set ''X'' equals the domain . Conversely, ''R'' is called right total if ''Y'' equals the range . When ''f'': ''X'' ...

. 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

as a one-one relation on the

s. Russell's series may be finite or generated by a relation giving

cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...

. In that case, the

point-pair separation In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering ''a b c d'' of four points has (''a,c'') and (''b,d'') as separating pairs. This point-pair s ...

relation is used for description. To define a progression, he requires the generating relation to be a

connected relation In mathematics, a relation on a set is called connected or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements. As described i ...

. Then

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...

s 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 Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...

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 "du ...

of measure. Informally, this parameter corresponds to

orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

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