Volume 1 / Part 4 — *152. DEFINITION AND ELEMENTARY PROPERTIES OF RELATION-NUMBERS.
Reading edition
*152. DEFINITION AND ELEMENTARY PROPERTIES OF RELATION-NUMBERS.
Summary of *152.
The relation-number of , which we denote by , is defined as the class of relations which are ordinally similar to , i.e. Hence our definition is The class of relation-numbers consists of all such classes as , i.e. These two definitions are analogous to those of *100, merely substituting "" for "." They are justified by similar considerations, and lead to similar results. With the exception of *152·7·71·72, the propositions of this number are the analogues of those of *100, and call for no remarks other than those in the introduction to *100 (mutatis mutandis).
*152·7·71·72 give relations between relation-numbers and cardinals. *152·7, which is constantly used, states that the cardinal number of consists of the fields of the relation-number of , i.e. the classes similar to are the fields of the relations similar to ; in symbols,
*152·7.
Hence it follows that the fields of a relation-number form a cardinal number, i.e.
*152·71.
Hence also it follows that cardinals other than consist of classes of the form , where is a relation-number other than , i.e.
*152·72.
In *154·9, we shall show how to remove the restriction to numbers other than , thus arriving at
*152·01.
*152·02.
*152·1.
*152·11.
*152·2.
*152·21.
*152·22.
*152·3.
*152·31.
*152·32.
*152·321.
*152·33.
Dem.
*152·35.
Dem.
In the above proposition, the same remarks as to types are to be made as in the case of *100·35. If in a certain type and are both null, we have in that type , but we need not have . Thus for example we shall find that, in the type of , But we do not have
*152·4.
Note that "," like "," is a formal number, and may be subjected to the conventions , , .
*152·41.
*152·42.
*152·43.
*152·44.
*152·45.
*152·5.
*152·51.
Dem.
*152·52.
The restriction involved in is, as we shall see later, not necessary, since in any assigned type.
*152·53.
Dem.
*152·54.
*152·6.
*152·62.
*152·63.
The utility of *152·6·62·63 is that they enable us to raise the type of a relation-number to any required extent. Thus gives a relation whose field is a class of the next type above that of , i.e. of the type ; while gives a relation whose field is , which is of the type . If , or, more generally, if , this is the type . Thus if we put , we have Thus is a relation whose field consists of terms of the same type as .
The following propositions on the relations of cardinals and relation-numbers are very important.
*152·7.
Dem.
*152·71.
*152·72.
Dem.
We shall show in *154·9 that the exclusion of in *152·72 is unnecessary.