Volume 1 / Part 3 — *102. ON CARDINAL NUMBERS OF ASSIGNED TYPES.

*102. ON CARDINAL NUMBERS OF ASSIGNED TYPES.

Summary of *102.

In this number, we shall consider a typically definite relation "," i.e. we shall consider the relation, to a class which is given as of the same type as , of the class of those classes which are similar to and of the same type as . We shall then put and the class of all such numbers as for a given and we shall call , so that

The notations here introduced for giving typical definiteness to "" and "" are those defined in *65 for any typically ambiguous relation.

By *63·01·02 we have, if is a typically ambiguous symbol,

Thus . If we apply the definitions to 1, "" is meaningless unless is a class; we therefore write a Greek letter in place of , and we have

If , we shall have . Hence

The converse implication also holds, so that

Thus consists of all unit classes whose sole members either are or are not members of , i.e. for which "" is significant.

In "," the hypothesis renders explicit the condition of significance; thus "" is always true when significant, and always significant when . On the interpretation of negative statements concerning types, see the note at the end of this number.

It should be noted that all the constant relations introduced in this work are typically ambiguous. Consider e.g. , , , , , , , , , . These all have more or less typical ambiguity, though all of them have what we will call relative typical definiteness, i.e. when the type of the relatum is given, that of the referent is given also. (In regard to , it is not true that, conversely, when the type of the referent is given, that of the relatum is also given.) But "" and "" have not even relative definiteness. When the type of the relatum is given, that of the referent becomes no more definite than before; the only restrictions are that the relatum for "" or "" must be a class, that the referent for "" must be a class, and that the referent for "" must be a class of classes. When a relation has relative definiteness, it is enough to fix the type of the relatum; and if further , so that leads to a descriptive function, "" has complete typical definiteness as soon as the type of is given. Now the constant relations hitherto introduced, with the exception of "" and "," have all been one-many relations, and have been used almost exclusively in the form of descriptive functions. Hence no special notation has been required to give typical definiteness, since "" in these circumstances, has typical definiteness as soon as is assigned. But with the consideration of "" and "," which do not have even relative definiteness, an explicit means of giving typical definiteness becomes necessary. It should be observed, however, that "" has typical definiteness, when is known, as soon as the domain of "" has typical definiteness, since must belong to the converse domain. It is for the sake of this and similar cases that we introduced the two definitions in *65, which only give typical definiteness to the domain.

In virtue of the definitions in *65, if is a typically ambiguous relation, and is a referent, becomes ; if, further, is a relatum, becomes . If is a referent for , we have , and . Thus has a member of the type next above that of , i.e. of the type of . Thus as was proved in *65. Hence in particular

It is chiefly for this reason that it is worth while to introduce the definition of .

We have, in virtue of the above, as will be proved in *102·46,

With regard to "," which is to be interpreted by *65·04, some caution is necessary. This will mean some one of those typically different relations called "" which have their domains composed of terms of the same type as . But it will not mean the logical sum of all such relations, because these relations are of different types according as their converse domains differ in type, and therefore their logical sum is meaningless. Thus for example if the type of is lower than or equal to that of , we shall have whence, if "" has its converse domain composed of terms of the same type as , But if is of higher type than , we shall find Thus "" is indeterminate in a way that makes a practical difference.

Exactly similar remarks apply to . We have thus "" shares the ambiguity of "." The question whether depends upon the decision of this ambiguity. The difficulty is that "" stands for the domain of any one determination of "" which has its domain composed of objects of the type of ; but it is the domain of only one such determination of "," because different determinations are of different types, and therefore cannot be taken together, even when their domains are all of the same type. In consequence of this ambiguity, "" is a symbol which is as a rule better avoided, and "" is not often useful except as a descriptive function, in which case the relatum supplies the requisite typical definiteness.

The peculiarity of "" is that it is typically definite, and yet is capable of different meanings: it is not wholly definite, being defined as the domain of a relation whose converse domain is typically ambiguous. It results that we cannot profitably make "" half-definite, as "" does, but must make it completely definite, as we do by taking . For this we adopt the notation . We cannot adopt the notation , because that would conflict with *65·11, nor , because that would conflict with *65·01, nor , for the same reason. But has no previously defined meaning. We may if we like regard "" as . Then the required meaning of "" would result from *65·04. But as "" so defined is not required, it is simpler to regard "" as a single symbol. We therefore put

*102·01.

The present number begins with various propositions (*102·2—·27) on a typically definite relation of similarity, i.e. . We then have a set of propositions (*102·3-·46) on "." This is only significant if and are of the same type; it then denotes the class of those classes which are similar to and of the same type as . We then have a set of propositions (*102·5-·64) on , i.e. on cardinals consisting of classes of the same type as which are similar to classes of the same type as . We next prove (*102·71—·75) that no sub-class of \alpha is similar to , and therefore (substituting for ) no class of the same type as is similar to , and therefore

*102*74.

This proves that is a cardinal, which is a proposition constantly required. The remaining propositions of *102 are concerned with where is a typically definite cardinal.

The most useful propositions in this number (apart from *102·74) are

*102·3.

*102·46.

*102·5.

*102·6.

*102·72.

This is used in proving , which is the proposition from which Cantor deduced that there is no greatest cardinal. (If , , and thus there is a rise of type.)

*102·84.

*102·85.

*102·01.

*102·11.

Here, if is a real variable, the conditions of significance require . But if is a typically ambiguous constant, such as or or , is a typically definite constant. It is chiefly for such cases that propositions such as the above are useful.

Dem.

*102·13.

*102·2.

*102·21.

*102·22.

102·23.

*102·24.

Dem.

*102·25.

*102·26.

Dem.

*102·27.

*102·3.

Dem.

*102·31.

Dem.

*102·32.

Dem.

*102·34.

*102·35.

*102·36.

This proposition is true whenever it is significant, and is significant whenever . When belongs to some other type, the above proposition is not significant.

*102·361.

*102·37.

Dem.

*102·4.

*102·41.

*102·42.

*102·43.

This inference is legitimate because, when is given, "" is typically definite. The inference from "" (which is true) to "" is not valid, because "" may hold only for some of the possible determinations of the ambiguity of "."

*102·44.

Dem.

*102·45.

Dem.

*102·46.

*102·5.

In using propositions, such as those of *100, in which we have a typically ambiguous "" or "," any significant typical definiteness may be added, since, when a typically ambiguous proposition is asserted, that includes the assertion of every possible proposition resulting from determining the ambiguity.

*102·501.

*102·51.

Dem.

*102·52.

*102·53.

Dem.

*102·54.

*102·541.

Dem.

*102·55.

Dem.

The above proposition shows that, if every class of the same type as is similar to some class of the same type as , then, given a class of the same type as , there is a class , of the same type as , such that the classes similar to and of the same type as are the same as the classes similar to and of the same type as ; and conversely, given any class , of the same type as , and similar to some class of the same type as , then there is a class , of the same type as , such that the classes similar to and of the same type as are the same as the classes similar to and of the same type as . We may express this by saying that, if the cardinals which go from the type of to the type of are never null, then those that go from the type of to the type of , with the exception of (if is one of them), are the same as those that begin and end within the type of . The latter are what we call "homogeneous" cardinals. Thus our proposition is a step towards reducing the general study of cardinals to that of homogeneous cardinals.

*102·6.

Dem.

*102·61.

Dem.

*102·62.

Dem.

*102·63.

Dem.

*102·64.

The following propositions are part of Cantor's proof that there is no greatest cardinal. They are inserted here in order to enable us to prove that is a cardinal, namely what we call a "descending" cardinal, i.e. one whose corresponding "" goes from a higher to a lower type.

*102·71.

Dem.

*102·72.

Dem.

*102·73.

Dem.

This proposition proves that no class of the same type as is similar to . Now is the greatest class of its type; thus there are classes of the type next above that of which are too great to be similar to any class of the type of . Thus (as will be explicitly proved later) the maximum cardinal in one type is less than that in the next higher type. Cantor's proposition that there is no maximum cardinal only holds when we are allowed to rise to continually higher types: in each type, there is a maximum for that type, namely the number of members of the type.

*102·74.

Dem.

*102·75.

Dem.

*102·8.

Dem.

*102·81.

Dem.

*102·82.

*102·83.

Dem.

*102·84.

Dem.

*102·85.

*102·86.

Dem.

*102·861.

Dem.

*102·862.

Dem.

*102·863.

Dem.

*102·87.

*102·88.

Dem.

Note on negative statements concerning types. Statements such as "" or "" are always false when they are significant. Hence when an object belongs to one type, there is no significant way of expressing what we mean when we say that it does not belong to some other type. The reason is that, when, for example, and are said to be different, the statement is only significant if interpreted as applying to the symbols, i.e. as meaning to deny that the two symbols denote the same class. We cannot assert that they denote different classes, since "" is not significant, but we can deny that they denote the same class. Owing to this peculiarity, propositions dealing with types acquire their importance largely from the fact that they can be interpreted as dealing with the symbols rather than directly with the objects denoted by the symbols. Another reason for the importance of typically definite propositions is that, when they are implications of which the hypothesis can be asserted, they can be used for inference, i.e. for the assertion of the conclusion. Where typically ambiguous symbols occur in implications, on the contrary, the conditions of significance may be different for the hypothesis and the conclusion, so that fallacies may arise from the use of such implications in inference. E.g. it is a fallacy to infer "" from the (true) propositions "" and "." (The truth of the first of these two requires that "" should receive the same typical determination in both its occurrences.) For these two reasons hypothetical concerning types are often useful, in spite of the fact that their hypotheses are always true when they are significant.