Volume 1 / Part 3 — *124. REFLEXIVE CLASSES AND CARDINALS.

*124. REFLEXIVE CLASSES AND CARDINALS.

Summary of *124.

In this number, we have to take up the second definition of infinity mentioned in the introduction to this Section. A class which is infinite according to this definition we propose to call a reflexive class, because a class which is of this kind is capable of reflexion into a part of itself. A class is called reflexive when there is a one-one relation which correlates the class with a proper part of itself. (A proper part is a part not the whole.) A reflexive cardinal is the homogeneous cardinal of a reflexive class.

We prove easily that reflexive classes are not inductive (*124·271), that reflexive cardinals are such as are greater than or equal to (*124·23), and such as are unchanged by adding 1 (excepting ) (*124·25). To prove that classes which are not inductive must be reflexive has not hitherto been found possible without assuming the multiplicative axiom. We do not need, however, to assume the axiom generally, but only as applied to products of factors. With this assumption, the result follows by a series of propositions explained below. Thus if a product of factors, no one of which is zero, is never zero, then the two definitions of the finite and the infinite coincide (*124·56).

We will call a cardinal a "multiplicative cardinal" if a product of factors none of which are zero is never zero. Thus all inductive cardinals are multiplicative cardinals; and the assumption needed for identifying the two definitions of finite and infinite is that should be a multiplicative cardinal.

For a reflexive class we use the notation "," and for a reflexive cardinal we use "." We define a reflexive cardinal as the homogeneous cardinal of a reflexive class, i.e. we put The only effect of this is to exclude from reflexive cardinals, which is convenient. We then need (on the analogy of *110·03*110·04) a definition of what is meant when an ambiguous symbol such as is said to be reflexive, and we therefore put

For the class of multiplicative cardinals we use the notation "." Thus we put whence it follows that if , a product of factors, none of which is zero, will never be zero.

We begin, in this number, with the more obvious properties of , proving that a is one which contains sub-classes of terms (*124·15), that it is one whose number is unchanged when a single term is taken away (*124·17), and that it remains reflexive if any inductive class is taken away from it (*124·182).

We then give corresponding propositions concerning (*124·23·25·252), proving, in addition to propositions already mentioned, that a reflexive cardinal is greater than every inductive cardinal (*124·26), and that a class which is neither inductive nor reflexive (if there be such) is one which neither contains nor is contained in any progression (*124·34). On such classes, see the remarks at the end of this number.

We then (*124·4·41) give a proposition merely embodying the definition of , and show that all inductive cardinals are multiplicative, which follows immediately from *120·62.

The following series of propositions (*124·51 ff.) are concerned with the proof that, if is a multiplicative cardinal, then the two definitions of finite and infinite coalesce. The proof, which is somewhat complicated, proceeds as follows.

To begin with, we know that if is a class which is not inductive, it contains classes having terms, if is any inductive cardinal. Thus we have The classes of classes , , ... , ... thus form a progression, which is contained in . Hence (*124·511) So far, the multiplicative axiom is not required.

The above progression of classes of classes is If is a selective relation for this class of classes, is a progression contained in . Hence

*124·513.

whence

*124·514.

To prove the next step, namely we make a fresh start. We have, by hypothesis, a progression whose domain is contained in ; hence . Thus it will suffice to prove where the conditions of significance require that should consist of classes.

For this purpose, we prove that no member of can be the last that has new members which have not occurred before. The proof proceeds by showing that if this were not so, would be an inductive class, and therefore, by *120·75, would be an inductive class. Hence (*124·534) the members of which introduce new terms form an , by *123·19; and so therefore do the classes of new terms which they introduce (*124·535). Hence (*124·536) a selection from these classes of new terms, which is a sub-class of , is also an , and therefore (*124·54) there is a progression contained in if the selection in question exists. This completes the proof.

In virtue of *124·511 and *120·74, we have, without the multiplicative axiom,

*124·6.

Hence if it could be shown that cannot be reflexive unless is reflexive, a double application of this would enable us, by means of *124·6, to identify the two definitions of the finite without the multiplicative axiom.

*124·01.

An equivalent definition would be

*124·02.

*124·021.

*124·03.

*124·1.

*124·11.

*124·12.

*124·13.

*124·14.

Dem.

*124·141.

Dem.

*124·15.

Dem.

*124·151.

*124·16.

Dem.

*124·17.

Dem.

*124·18.

*124·181.

Dem.

*124·182.

*124·2.

*124·21.

*124·23.

Dem.

*124·231.

*124·232.

*124·24.

Dem.

*124·25.

*124·251.

*124·252.

Dem.

*124·253.

Dem.

*124·26.

Dem.

*124·27.

*124·271.

Dem.

*124·28.

Dem.

*124·29.

Dem.

*124·3.

*124·31.

In virtue of the above proposition, if there are any numbers which are neither inductive nor reflexive, they are such as are neither greater than, less than, nor equal to . (The existence of in a suitable type can be deduced from the existence of numbers which are neither inductive nor reflexive; cf. *124·6.) Two further propositions (*124·33·34) are given below on non-inductive non-reflexive classes and cardinals. The subject is resumed in the remarks at the end of the number.

*124·33.

*124·34.

Dem.

*124·4.

*124·41.

The following propositions give the proof of *124·56, which identifies the two definitions of the finite, on the assumption that is a multiplicative cardinal. (*124·513, however, is only used in proving *124·514, and *124·514 is not used in the proof. It is retained as marking a stage in the argument, although the actual propositions subsequently used are not it, but the lemmas which lead to it.)

*124·51.

here has the meaning defined in *123·02.

Dem.

*124·511.

*124·512.

Dem.

*124·513.

*124·514.

The following propositions are concerned in proving that, if is a multiplicative cardinal, then a class such as in *124·512 must be such that a progression is contained in . The characteristics of which are used in the proof are . Since , we have . Hence the hypothesis with which the following series of propositions is concerned is but the earlier propositions do not need the full hypothesis.

In what follows, note that if is the class of those terms which occur in and have never occurred before in any earlier member of . We prove that, with our hypothesis, members of for which this class of new terms is not null form a class which has no last member, and therefore form a progression.

*124·52.

Dem.

*124·521.

Dem.

*124·53.

*124*531.

Dem.

*124·532.

*124·533.

Dem.

*124·534.

Dem.

*124·535.

*124·536.

Dem.

*124·54.

Dem.

*124·541.

Dem.

*124·55.

Dem.

*124·56.

Dem.

The above proposition identifies the two definitions of the finite, on the hypothesis .

*124·57.

*124·58.

Dem.

The above proposition gives another hypothesis which would enable us to identify the two definitions of the finite if it could be proved, namely or, what comes to the same thing,

*124·6.

Dem.

*124·61.

Dem.

The following properties of cardinals which are neither inductive nor reflexive (supposing there are such) are easily proved. Let us put

where "" stands for "mediate." Then Hence mediate cardinals have no maximum or minimum. whence since we have either or .