Volume 1 / Part 3 — *122. PROGRESSIONS.

*122. PROGRESSIONS.

Summary of *122.

By a "progression" we mean a series which is like the series of the inductive cardinals in order of magnitude (assuming that all inductive cardinals exist), i.e. a series whose terms can be called where every term of the series is correlated with some inductive cardinal, and every inductive cardinal is correlated with some term of the series. Such series belong to the relation-number (cf. *152 and *263) which Cantor calls . Their generating relation may be taken to be the transitive relation of earlier and later, or the one-one relation of immediate predecessor to immediate successor. We shall reserve the notation for the transitive generating relations of progressions; for the present, we are concerned with the one-one relations which generate progressions. The class of these relations we shall call "."

It is not convenient to define a progression as a series which is ordinally similar to that of the inductive cardinals, both because this definition only applies if we assume the axiom of infinity, and because we have in any case to show that (assuming the axiom of infinity) the series of inductive cardinals has certain properties, which can be used to afford a direct definition of progressions. The existence of progressions, however, is only obtainable by means of the axiom of infinity, and is then most easily obtained from the fact that the inductive cardinals form a progression. We shall not consider the existence-theorem until the next number (*123).

From this number onwards convention of the Prefatory Statement is used when relevant.

The characteristics of the generating relation of a progression, which we employ in the definition, are the following:

(1) is a one-one relation;

(2) there is a first term, i.e. ;

(3) the whole field is contained in the posterity of the first term, i.e. . (If this failed, would consist of two or more distinct families, of which, since we have , all but one would have to be cyclic families.)

(4) every term of the field has a successor, i.e. the series is endless. This is secured , or (what is equivalent) .

These four properties suffice to define the one-one generating relations of progressions. It will be observed that (2), (3) and (4) are all secured by

This secures , by *14·21; it secures , by *37·25 and *90·163; hence, by *33·181, , and therefore

Hence our definition of progressions is

Instead of stating in the definition that is to be a one-one relation, it is sufficient to put , which, with , implies , and may be substituted for without altering the force of the definition (*122·17).

In the present number we shall prove, among other propositions, that every existent class contained in a progression has a first term (*122·23), i.e. that progressions are well-ordered series; that in a progression (*122·16), which makes the propositions of *121 available; that if is any inductive cardinal other than 0, exists (*122·33), i.e. the series has a th term; that any class contained in and having a last term is an inductive class (*122·43), and that any class contained in and not having a last term is itself the domain of a progression (*122·45), so that every class contained in is either inductive or the domain of a progression (*122·46); that if is a many-one, and a member of its domain, and if the descendants of have no last term and are none of them descendants of themselves, then arranges these descendants in a progression (*122·51); and that the same holds if is a one-one and (*122·52); and that if and belongs to one of the generations of , but not to one of the generations of , then arranges the whole family of in a progression (*122·54).

The following general observations on the families of one-one relations may serve to elucidate the bearing of the propositions of this section.

Given any relation , we call , i.e. the family of . If is a one-one, this family may be of four different kinds. (1) It may be a closed series, like the angles of a polygon. This occurs if . In this case the family forms an inductive class. (2) It may be an open series with a beginning and an end; this occurs if

In this case also the family forms an inductive class. (3) It may be an open series with a beginning and no end, or an end and no beginning. This occurs if or if In this case, the series is of the type or , and is non-inductive and reflexive. (4) The series may be open and have neither beginning nor end. This occurs if In this case we get a series whose relation-number is the sum (in the sense of *180) of and , which again is non-inductive and reflexive. In all four cases, if and be any two members of the family of , the interval between and is an inductive class.

If is a member of , or if the family of contains a member of , cases (1) and (4) are excluded, since the series has a beginning. In this case the number of predecessors of any term is an inductive number. It will be observed that every family is either wholly contained in or wholly contained in ; families of kinds (2) and (3) (excluding, in (2), those which have an end but no beginning) are contained in , while families of kinds (1) and (4), and those of (2) which have an end but no beginning, are contained in ; families containing a member of are contained in , while all others are contained in .

Thus a one-one relation in general gives rise to a number of wholly disconnected series, some closed, others open and with or without a beginning or an end. The condition that all the series should be open is .

The case of a -shaped family, considered in *96, cannot arise when , for in a -shaped family the term at the junction of the tail and the circle has two predecessors, one in the tail and one in the circle, so that the relation in question is not . It follows that, when , if is a family containing a member of , (cf. *96·23).

When exists, there is only one family which has a beginning. In this case, ignoring the other families (if any), we call the members of the family of respectively , , , .... If the family has members, where is an inductive cardinal, its last member will be . If on the other hand the number of members of the family is not an inductive cardinal, it must be ; in this case, the family forms a progression, whose members are , , , ..., , ..., where always exists when is an inductive cardinal.

In addition to the propositions already mentioned, the following are important:

*122·21.

(Cf. note to *122·21, below.)

*122·34.

*122·341.

In virtue of these two propositions, the terms of a progression are where every inductive cardinal occurs. This is the same fact as is usually assumed when the terms are represented as

*122·35.

*122·36.

*122·37.

*122·38.

I.e. the number of terms up to any given point of a progression is inductive.

*122·01.

*122·1.

*122·11.

Dem.

Observe that, by the conventions as to descriptive symbols, involves the existence of , whereas does not, since, if does not exist, we have , and therefore will satisfy the equivalence, i.e. will satisfy the equivalence although it has no first term. This is the reason why appears explicitly in *122·11, though it was only implicit in *122·1.

*122·12.

*122·14.

Dem.

*122·141.

Dem.

*122·142.

*122·143.

*122·15.

Dem.

*122·151.

*122·152.

*122·16.

This proposition enables us to apply to progressions all the propositions of *121 in which we have as hypothesis

*122·17.

Dem.

To illustrate this proposition, consider its application to the inductive cardinals arranged in order of magnitude; i.e. take as a value of the relation

We then have ; also

We have also so that .

Again whence i.e.

But we do not get or unless we have which is the axiom of infinity. If this condition fails, we reach at last an inductive cardinal which = , and we have so that has two immediate predecessors, namely itself and the last existent cardinal. The posterity of 0, in this case, is a in which the circle has narrowed to a single term, namely .

Thus we need the axiom of infinity in order to prove

*122·2.

*122·21.

This proposition, together with *122·16 and *91·56, shows that if , has the three properties by which transitive serial relations are defined (cf. *204), namely it is (1) transitive, (2) contained in diversity, (3) connected, i.e. such that it relates any two distinct members of its field. We shall at a later stage define the ordinal number as the class of such relations as , where .

*122·22.

Dem.

*122·23.

Dem.

This proposition shows that every existent class contained in a progression has a first term, i.e. that a progression is a well-ordered series (cf. *250).

*122·231.

Dem.

*122·24.

Dem.

Except when , will not reduce to a single term. In fact, if , , i.e. consists of the first terms of the progression.

*122·25.

Dem.

The above proposition shows that what we may call an "arithmetical progression" in a progression is a progression, i.e. if, starting from any term of a progression, we take every other term, or every third term, or every th term, we still have a progression.

*122·26.

Dem.

The above proposition shows that if an existent class contained in a progression has no maximum, then any assigned member of the progression is succeeded by members of the class.

The following proposition states that if has members belonging to a progression, and there are members of the progression which do not precede any member of , then there is in the progression a last member of .

*122·27.

Dem.

*122·28.

Dem.

*122·3.

*122·31.

Dem.

*122·32.

Dem.

*122·33.

Dem.

*122·34.

*122·341.

Dem.

In virtue of *122·34·341, all the terms of a progression occur in the series , , ... , ..., and every inductive cardinal except 0 is used in forming this series.

*122·35.

Dem.

*122·36.

Dem.

*122·37.

Dem.

*122·38.

*122·381.

The following series of propositions are concerned in proving that any class contained in a progression is inductive if it has a last term, and is a progression if it has no last term. In the latter case, it is supposed arranged in the same order as it had in the original progression. A certain complication is necessary in order to define its one-one generating relation. If is the generating relation of the original progression, we proceed first to , then to , where is the class in question; this gives us a transitive generating relation for . Calling this relation , we then proceed to , i.e. the relation of consecutive members of the series generated by . This relation turns out to be one-one, and to arrange in a progression; hence our proposition is proved. The reason for the necessity of this detour is that consecutive members of may not be consecutive members of the original progression.

*122*41.

Dem.

*122·42.

Dem.

*122·43.

Thus every class which is contained in a progression and has a last term is inductive. We have next to prove This is effected in the following propositions.

*122·44.

Note. The hypothesis here exceeds what is necessary for the conclusion, but is the hypothesis required for *122·45, for which the present and the following propositions are lemmas.

Dem.

*122·441.

Dem.

*122*442.

In proving below, we assume and consider the maximum of , which is shown to exist and be , whence .

Dem.

*122·443.

Dem.

*122·444.

Dem.

*122·45.

This proposition shows that every series extracted from a progression and having no last term is a progression.

*122·46.

This proposition shows that any number less than the number of terms in a progression is inductive. This result will be developed in the next number (*123).

*122·47.

Dem.

*122·48.

Dem.

*122·49.

The following propositions are concerned with circumstances under which the posterity or the family of a term forms a progression.

*122·51.

Here has the meaning defined in *96.

Dem.

The following proposition (*122·52) is used in *123·191, *261·4 and *264·22.

*122·52.

Dem.

The remaining propositions (*122·53·54·55) are not used in the sequel.

*122·53.

Dem.

*122·54.

Dem.

*122·55.

Dem.