set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...

a serial relation is a

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''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 cal ...

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 to define

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

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 philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...

used serial relations in ''

The Principles of Mathematics ''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented Russell's paradox, his famous paradox and argued his thesis that mathematics and logic are identical. The book presents a view of ...

'' (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 ...

, irreflexivity, and transitivity. A serial relation ''R'' is an endorelation 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. 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 In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed ...

, 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 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 ...

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

using

scientific notation Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits. It may be referred to as scientif ...

, where a power of ten represents a

decade A decade (from , , ) is a period of 10 years. Decades may describe any 10-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 ...

of measure. Informally, this parameter corresponds to

orders of magnitude In a ratio scale based on powers of ten, the order of magnitude is a measure of the nearness of two figures. Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are wi ...

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. Properties of binary relations Order theory