Volume 1 / Part 3 — *103. HOMOGENEOUS CARDINALS.
Reading edition
*103. HOMOGENEOUS CARDINALS.
Summary of *103.
In this number, we shall consider cardinals generated by a homogeneous relation of similarity. A "homogeneous" cardinal is to mean all the classes similar to some class and of the same type as . The "homogeneous cardinal of " will be defined as ; we shall denote it by "." Then the class of homogeneous cardinals is the class of all such cardinals as "," i.e. it is ; this we shall denote by "." The symbol "" is typically definite as soon as is assigned; "," on the contrary, is typically ambiguous: it must be a , but otherwise its type may vary indefinitely. Homogeneous cardinals have, however, many properties which do not require that the ambiguity of "" should be determined, and few which do require this. They are important also as being the simplest kind of cardinals, and as being a kind to which other kinds can usually be reduced.
The chief advantage of homogeneous cardinals is that they are never null (*103·13·22). This enables us to avoid by their means the explicit exclusion of exceptional cases; thus throughout Section B we shall use homogeneous cardinals in defining the arithmetical operations: the arithmetical sum of and , for example, will be defined by means of and , in order to exclude such a determination of the typical ambiguity of and as would make either of them null. It is true that not only homogeneous cardinals, but also ascending cardinals (cf. *104), are never null. But homogeneous cardinals are much the simplest kind of cardinals that are never null, and are therefore the most convenient.
The fact that no homogeneous cardinal is null is derived from
*103·12.
Other important propositions in this number are the following:
*103·2.
*103·26.
The above proposition is used constantly.
*103·27.
Thus to say that is the homogeneous cardinal of is equivalent to saying that is a cardinal of which is a member.
*103·301.
*103·34.
*103·4.
*103·41.
*103·01.
*103·02.
*103·1.
*103·11.
*103·12.
*103·13.
This is a legitimate inference from *103·12 because, when is given, is typically definite.
*103·14.
Dem.
*103·15.
Dem.
*103·16.
In this proposition, the equation "" must be supposed to hold in any type for which it is significant. Otherwise, we might find a type for which , without having .
Dem.
*103·2.
*103·21.
In adducing a proposition, such as *100·2, which is concerned with an "" entirely undetermined in type, any degree of typical determination may be added to our "," since an asserted proposition containing an ambiguous "" is only legitimate if it is true for every possible determination of the ambiguity.
*103·22.
*103·23.
*103·24.
*103·25.
*103·26.
Dem.
*103·27.
Dem.
*103·28.
Dem.
*103·3.
Dem.
*103·301.
Note that although "" is not definite, "" is absolutely definite as soon as is assigned.
Dem.
*103·31.
Dem.
*103·32.
Dem.
In the above proposition, the "" may be omitted, and we may write (cf. *103·33, below)
For the is wholly arbitrary, so that any possible determination of makes the above proposition true. We may proceed a step further, and write (*103·34, below)
But although we also have , provided the "" on the right is suitably determined, we do not have this always. For example, if "" is determined as , and "" as , then .
*103·33.
Dem.
*103·34.
Dem.
Thus every cardinal except is a homogeneous cardinal in the appropriate type. Note that although of course every homogeneous cardinal is a cardinal, yet "" must not be asserted, because it is possible to determine the ambiguity of "" in such a way as to make this false. Hence we do not get .
*103·35.
The hypothesis of this proposition is satisfied, as will appear later, if the type of is in what we may call the direct ascent from that of , i.e. if it can be reached from by a finite number of steps each of which takes us from a type to either or . Thus in such a case the cardinals (other than ) which go from to are the same as those which begin and end within . It will also appear that in such a case always is a member of . If two cardinals which are not equal must always be one greater and the other less, then is the condition for . In that case, we shall have . But there is no known proof that of two different cardinals one must be the greater, except by assuming the multiplicative axiom and proving thence (by Zermelo's theorem) that every class can be well-ordered (cf. *258).
*103·4.
Dem.
*103·41.
Dem.
*103·42.
Dem.
*103·43.
Dem.
*103·44.
Dem.
*103.5.
Dem.
*103·51.
Dem.
0 and 1 are the only cardinals of which the above property can be proved universally with our assumptions. If (as is possible so far as our assumptions go) the lowest type is a unit class, we shall have in that type (though in no other) , so that in that type .