Volume 1 / Part 3 — *110. THE ARITHMETICAL SUM OF TWO CLASSES AND OF TWO CARDINALS.
Reading edition
*110. THE ARITHMETICAL SUM OF TWO CLASSES AND OF TWO CARDINALS.
Summary of *110.
In this number, we start from the definition:
*110·01.
is called the "arithmetical class-sum" of and . The definition is framed so as to give two mutually exclusive classes respectively similar to and , so that the number of terms in the logical sum of these two classes is the arithmetical sum of the numbers of terms in and respectively. is significant whenever and are classes, whatever their types may be.
By means of , we define the arithmetical sum of two cardinals as follows:
*110·02.
This defines the "arithmetical sum of two cardinals." (It is not necessary to significance that and should be cardinals, but only that they should be classes of classes. If, however, either is not a cardinal, ). It will be observed that, when and are typically definite, so are and in the above definition; but is typically ambiguous, on account of the ambiguity of "." Hence is also typically ambiguous.
It will be shown that is always a cardinal, and that, if Hence whenever and are cardinals other than , is an existent cardinal in some types, though it may be in others.
Two more definitions are required in this number, namely:
*110·03.
*110·04.
These definitions are needed in order to apply the definition of to the case in which and are replaced by typically ambiguous symbols and . It does not make any difference to the value of how the ambiguities of and are determined, so long as they are determined in a way that insures ; but if there are types in which either or is , we get in all types if we determine the ambiguities so that or . It is in order to exclude such determinations of the ambiguity that the above definitions are required. Also in connection with these definitions and the corresponding definitions *113·04·05 and *116·03·04 and *117·02·03, the convention of the prefatory statement must be noted.
The propositions of the present number begin with the properties of . We show (*110·11·12) that consists of two mutually exclusive parts, which are respectively similar to and ; we show (*110·14) that if and are mutually exclusive, is similar to , and (*110·15) that if and are respectively similar to and , then is similar to . We show (*110·16) that consists of all classes which can be divided into two mutually exclusive parts which are respectively similar to and .
We then proceed (*110·2—·252) to the consideration of . Here and are typically definite, and the definition *110·02 applies to any typically definite symbols, such as or . We prove (*110·21) that if and are cardinals, their sum consists of all classes similar to some class of the form , where ; we prove (*110·22) that the sum of and is , and (*110·25) that if and are cardinals, their sum is equal to the sum of the "same" cardinals in any other types in which they are not null, i.e.
*110·25.
We then (*110·3—·351) consider , to which we apply the definitions *110·03·04. We have
*110·3. whence the other properties of follow from previous propositions.
We then have (*110·4—·44) various propositions on the type of and its existence and kindred matters. The chief of these are
*110·4.
*110·42.
This proposition requires no hypothesis, because, if and are not both cardinals, , and is a cardinal, by *102·74.
Our next set of propositions (*110·5—·57) are concerned with the permutative and associative laws, which are *110·51 and *110·56 respectively.
We then (*110·6—·643) consider the addition of 0 or 1, proving (*110·61) that a cardinal is unchanged by the addition of 0, and (*110·643) that .
*110·01.
*110·02.
*110·03.
*110·04.
These definitions are extended by of the prefatory statement.
*110·1.
*110·101.
Dem.
*110·11.
Dem.
*110·12.
*110·11·12 give the justification for the use of in defining arithmetical addition, since they show that consists of two mutually exclusive parts which are respectively similar to and .
*110·13.
Dem.
*110·14.
Thus whenever and are mutually exclusive, their logical sum may replace their arithmetical sum in defining the sum of their cardinal numbers.
*110·15.
Dem.
*110·151.
Dem.
*110·152.
Dem.
*110·16.
*110·17.
Dem.
Thus when and are of the same type, exists at least in the type next above that of and . We cannot prove that it exists in the type of and . E.g. suppose the lowest type contained only one member; then if were that one member, would not exist in the type to which belongs, but would exist in the next type, i.e. there would not be two individuals, but there would be two classes, namely and , so that .
*110·18.
Dem.
*110·2.
*110·201.
*110·202.
Dem.
*110·21.
*110·211.
Dem.
*110·212.
Dem.
*110·22.
Dem.
*110·221.
Dem.
*110·23.
Thus is independent of and so long as and exist in the types of and respectively.
*110·231.
*110·24.
Dem.
*110·25.
Dem.
*110·251.
Dem.
*110·252.
A similar proof applies to , , etc., and to any such derived cardinals whose existence follows from that of and . The proposition does not hold generally for , and other descending derived cardinals, because they may be null when and exist.
The following proposition (*110·3) is more often used than any other in this number except *110·4.
*110·3.
*110·31.
The following proposition is frequently used.
*110·32.
*110·33.
The above proposition is used in *110·63. We might have used the above to define arithmetical addition, but this method would have been less convenient than the method adopted in this number, both because there would have been more difficulty in dealing with types, and because the existence of (in the types in which it does exist) is less evident with the above definition than with the definitions given in *110·01·02·03·04.
*110·331.
Dem.
*110·34.
*110·35.
*110·351.
Similar propositions will hold generally for ascending cardinals.
The following proposition (*110·4) is the most used of the propositions in this number. It is useful both in the form given, and in the form resulting from transposition, in which it shows that unless both and are existent cardinals. It is chiefly useful in avoiding the necessity of the hypothesis , in such propositions as the commutative and associative laws.
*110·4.
The following propositions, down to *110·411 inclusive, are concerned with types. They are not referred to in the sequel.
*110·401.
Dem.
*110·402.
Dem.
*110·403.
*110·404.
*110·41.
Dem.
*110·411.
It will be observed that the following proposition (*110·42) requires no hypothesis. This is owing to *110·4 and *102·74.
*110·42.
Dem.
*110·43.
*110·44.
Dem.
The above proposition depends upon the fact that is typically ambiguous, even when and are typically definite. It is used in the theory of inductive cardinals (*120·32·41·424).
The following propositions are concerned with the commutative and associative laws for arithmetical addition of cardinals.
*110·5.
Dem.
*110·501.
*110·51.
It is not necessary to the truth of the above proposition that and should be cardinals. If either is not a cardinal, and are both .
The following propositions lead to the associative law (*110·56).
*110·52.
Dem.
*110·521.
*110·53.
*110·531.
*110·54.
Dem.
*110·541.
Dem.
*110·55.
*110·551.
*110·56.
Dem.
This is the associative law for arithmetical addition. It will be seen that, like the commutative law, it does not require that , , should be cardinals.
*110·561.
*110·57.
The following propositions, concerning the addition of 0 or 1, are used frequently in dealing with inductive cardinals (*120).
*110·6.
Dem.
When is a typically definite cardinal, is the same cardinal rendered typically ambiguous; when is a typically ambiguous cardinal, , is . In place of the above proposition, we might write ; this would be true whenever the ambiguity of was so determined as to make it significant. But the above form gives more information.
*110·61.
Dem.
In this proposition, is typically ambiguous; hence we escape the necessity of putting on the right, as we should have to do if were typically definite. We can deduce *110·61 from *110·6 as follows:
We have to travel via in this proof, in order to avoid the possibility of a typical determination of which would make . It is for the same reason that we cannot put ""; for if the first is determined to a type in which , while the second is not, this equation becomes false.
*110·62.
Dem.
*110·63.
Dem.
The above proposition is much used in the theory of finite and infinite, both cardinal and ordinal. It connects mathematical induction for inductive cardinals with mathematical induction for inductive classes (cf. *120).
*110·631.
Dem.
The proposition which might at first sight seem demonstrable, will only be true universally if the total number of objects in any one type is not finite. For suppose is a type, and . Then if is a finite class, . Hence . Hence in all types. But will exist in all types higher than that of . If on the other hand the number of entities in is infinite, we shall have Hence in this case the above proposition will be true universally.
*110·632.
Dem.
*110·64.
*110·641.
*110·642.
*110·643.
Dem.
The above proposition is occasionally useful. It is used at least three times, in *113·66 and *120·123·472.
*110·7·71 are required for proving *110·72, and *110·72 is used in *117·3, which is a fundamental proposition in the theory of greater and less.
*110·7.
Dem.
*110·71.
Dem.
The above proof depends upon the fact that "" and "" are typically ambiguous, and therefore, when they are asserted to be equal, this must hold in any type, and therefore, in particular, in that type for which we have , i.e. for . This is why the use of *100·3 is legitimate.
*110·72.
Dem.