Volume 1 / Part 4 — *155. HOMOGENEOUS RELATION-NUMBERS.
Reading edition
*155. HOMOGENEOUS RELATION-NUMBERS.
Summary of *155.
A relation-number is called homogeneous when it is generated by a homogeneous relation of likeness, i.e. when it consists of all relations which are like a given relation and of the same type as . For the homogeneous relation-number of we write ""; thus . When is given, is typically definite. We have always , hence . Conversely, if a typically definite relation-number is not null, it is a homogeneous relation-number; in fact, if is a member of it, it is . Thus the homogeneous relation-numbers are all the relation-numbers except .
Homogeneous relation-numbers play the same part in relation-arithmetic as homogeneous cardinals play in cardinal arithmetic. The propositions of this number (except *155·6·61) are the analogues of those with the same decimal part in *103. Their proofs are exactly analogous to the proofs of their analogues in *103, and are therefore omitted.
The following propositions are the most useful in this number.
*155·11.
This merely embodies the definition.
*155·12.
whence
*155·13.
*155·16.
This proposition is used in the theory of well-ordered series (*253 and *255). It requires that the equation "" on the right-hand side should be subject to the convention . Otherwise, the typical ambiguities might be so determined as to give , which would not imply .
*155·2.
This merely embodies the definition of .
*155·22.
*155·26.
*155·27.
*155·34.
*155·4.
*155·5.
*155·6.
This last proposition connects homogeneous relation-numbers with homogeneous cardinals.
*155·01.
*155·02.
*155·11.
*155·12.
*155·13.
*155·14.
*155·15.
*155·16.
*155·2.
*155·21.
*155·22.
*155·23.
*155·24.
*155·25.
*155*26.
*155*27.
*155·28.
*155·3.
*155·301.
*155·31.
*155·32.
*155·33.
*155·34.
*155·35.
*155·4.
*155·41.
*155·42.
*155·43.
*155·44.
*155·5.
*155·51.
*155·52.
The following propositions have no analogue in *103.
*155·6.
Dem.
*155·61.
On ascending and descending relation-numbers, propositions analogous to those of *104, *105, and *106 might be proved by proofs analogous to those given in those numbers. It is, however, scarcely necessary to add anything to the propositions already proved, namely *154·24·241·242·25·251 on descending relation-numbers, *154·26·261·262·31·311·32·321·322·33·331 on ascending relation-numbers, and *155·23·34 giving the relations of non-homogeneous to homogeneous relation-numbers. Ascending relation-numbers all exist, and those that start from the type of , wherever they end[13], are the correspondents[14] of the homogeneous relation-numbers of the type of , and are only some of the homogeneous relation-numbers of the type in which they end. Descending relation-numbers consist of together with the homogeneous relation-numbers of the type in which they end: they are the correspondents of only some of the type in which they begin, or rather, is the common correspondent of all those relation-numbers in the initial type which are not correspondents of any homogeneous relation-number in the end-type. These properties are exactly the same as in the case of cardinals, as might be foreseen by *154·14.