Volume 1 / Part 3 — *100. DEFINITION AND ELEMENTARY PROPERTIES OF CARDINAL NUMBERS.
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*100. DEFINITION AND ELEMENTARY PROPERTIES OF CARDINAL NUMBERS.
Summary of *100.
In this number we shall be concerned only with such immediate consequences of the definition of cardinal numbers as do not require typical definiteness, beyond what the inherent conditions of significance may bestow. We introduce here the fundamental definitions:
*100·01.
*100·02.
The definition "" is required chiefly for the sake of the descriptive function . We have
*100·1.
This may be stated in various equivalent forms, which are given at the beginning of this number (*100·1—·16). After a few propositions on as a relation, we proceed to the elementary properties of . We have
*100·3.
*100·31.
*100·321.
*100·33.
We proceed next to the elementary properties of . We have
*100·4.
*100·42.
*100·45.
*100·51.
Observe that when we have such a hypothesis as "," the , though it may be of any type, must be of some type; hence the cannot have the typical ambiguity which belongs to . If we put , this will hold only in the type of ; but "" is a typically ambiguous symbol, which will represent in any type the "same" number as . Thus "" is an equation which is applicable to all possible typical determinations of "" and "."
*100·52.
The hypothesis is unnecessary, but we cannot prove this till later (*102).
We end the number with some propositions (*100·6—·64) stating that various classes (such as ), which have already been proved to be similar to , have members.
*100·01.
*100·02.
*100·1.
*100·11.
*100·12.
*100·13.
Dem.
*100·14.
*100·15.
Dem.
*100·16.
Dem.
*100·2.
*100·21.
Dem.
*100·22.
*100·3.
Note that it is fallacious to infer , for reasons explained in the introduction to the present section.
*100·31.
*100·32.
*100·321.
Dem.
Note that is not always true. We might be tempted to prove it as follows:
But the use of *10·1 here is only legitimate when the "" concerned is a homogeneous relation. If , are descending cardinals, we may have without having .
*100·33.
Dem.
Note that we do not always have
For if the concerned is a descending , and and are sufficiently great, and may both be . For example, we have
But , so that
Thus "" is not always true when it is significant.
*100·34.
*100·35.
Dem.
Thus the only case in which the implications in *100·321·33·34 cannot be turned into equivalences is the case in which and are both .
*100·36.
*100·4.
*100·41.
*100·42.
Dem.
*100·43.
*100·44.
Dem.
*100·45.
*100·5.
Dem.
*100·51.
Dem.
*100·511.
Here the last "" may be of a different type from the others: the proposition holds however its type is determined.
Dem.
*100·52.
This proposition still holds when , but the proof is more difficult, since it depends upon the proof that every null-class of classes is an , which in turn depends upon the proof that is not similar to or to any class contained in .
*100·521.
Dem.
*100·53.
Dem.
*100·6.
*100·61.
*100·62.
*100·621.
*100·63.
*100·631.
*100·64.
Dem.