Volume 1 / Part 3 — *120. INDUCTIVE CARDINALS.

*120. INDUCTIVE CARDINALS.

Summary of *120.

Inductive Cardinals are those that obey mathematical induction starting from 0, i.e. in the language of Part II, Section E, they are the posterity of 0 with respect to the relation of to , or, in more popular language, they are those that can be reached from 0 by successive additions of 1. In former days, these were supposed to be all the cardinals, and mathematical induction was treated as a kind of self-evident axiom. We now know that only certain cardinals obey mathematical induction starting from 0. It is these cardinals which are to be considered in this number. They embrace 0, 1, 2, ... and generally all those cardinals which would be commonly called finite, all those which can be expressed in the usual Arabic system of numeration, and no others. The propositions to be proved concerning them in this number are elementary and familiar; the interest lies entirely in the definition and method of proof, not in the propositions themselves.

Put

Since has necessarily its domain and converse domain of the same type, it is important to be careful in noting the relations of type. Accordingly we also put

We begin by applying the propositions of *90. Thus we have

*120·11.

*120·12.

*120·121.

*120·13.

*120·15.

*120·151.

*120·152.

We then proceed to deduce the elementary properties of inductive classes, putting

We have

*120·21.

*120·211.

(We do not have an equivalence here, because, for aught we know, it might be possible to determine the ambiguity of so that , even when . his will not be possible, however, if the axiom of infinity is assumed.)

*120·212·213.

*120·214.

We have a set of propositions applying induction to classes directly, and not through the intermediary of cardinals. Thus we have

*120·251.

*120·26.

We then state the axiom of infinity, and prove (*120·33) that it is equivalent to the assumption that if is an inductive cardinal, . To prove this, we first prove various propositions about , among others the following:

*120·311.

*120·322.

We then proceed to consider subtraction (*120·41—·418), which only gives a cardinal number when the subtrahend is an inductive cardinal. We have

*120·41.

We might validly put instead of , since will be true whenever it is significant.

We have

*120·411.

*120·4111.

Hence we arrive at the conditions requisite for the usual point of view of subtraction; namely,

*120·412.

Also from *120·4111 we deduce

*120·414.

And from *120·411.*119·34, we find

*120·416.

We prove next that no proper part of an inductive class is similar to the whole (*120·426), i.e. that inductive classes are non-reflexive, and various connected propositions, e.g.

*120·423.

*120·4232.

*120·428.

*120·429.

The last two of the above propositions do not hold in general when is a cardinal which is not inductive.

We prove next that if is an existent inductive cardinal, then any existent cardinal is greater than, equal to, or less than (*120·441); that if , are inductive cardinals, so is (*120·45·4501), and if is an inductive cardinal other than , so are and (*120·452). We then have some propositions dealing with mathematical induction starting from 1 or 2, e.g.

*120·4622.

*120·47.

From *120·452 we deduce

*120·48.

so that any number less than an inductive number is inductive. Hence

*120·481.

which is a proposition constantly used, and

*120·491.

We then prove that if , are inductive cardinals, and are either inductive cardinals or (*120·5*120·52), while conversely if or is an existent inductive cardinal, and are so also, with exceptions for 0 and 1 (*120·512·56·561). Hence we infer the uniqueness of division and the taking of roots (*120·51·53·55) so long as inductive numbers are concerned.

We have next a set of propositions on the axiom of infinity and the multiplicative axiom. We prove (*120·61) that if there is any existent cardinal which is not inductive, the axiom of infinity is true. From *83·9·904, we infer by induction that if is an inductive class of which is not a number, exists (*120·62), whence it follows that either the multiplicative axiom or the axiom of infinity must be true (*120·64).

Finally, we have a set of propositions on inductive classes. We prove

*120·71.

*120·74.

*120·75.

with analogous propositions (involving however a hypothesis as to ) on the subject of .

The propositions of the present number are essential to the ordinary arithmetic of finite numbers. In the present work, however, they are not much used after the present section until we reach Part V, Section E, where we deal with the ordinal theory of finite and infinite.

*120·01.

Note that in virtue of our general conventions for descriptive functions of two arguments (*38), That is, is the relation of a cardinal to its immediate predecessor. It is the number written in the usual mathematical notation as +1 in the series of positive and negative integers, just as its converse is the number -1. (It should be observed that if is any cardinal, + is not identical with , since + is a relation, while is a class of classes.)

*120·011.

All members of belong to the same type as , so that, if is any member of , "" is significant.

*120·02.

*120·021.

In virtue of these definitions an inductive class is one whose cardinal is an inductive cardinal.

*120·03.

"," like "," is an arithmetical hypothesis which some will consider self-evident, but which we prefer to keep as a hypothesis, and to adduce in that form whenever it is relevant. Like "," it states an existence-theorem. In the above form, it states that, if is any inductive cardinal, there is at least one class (of the type in question) which has terms. An equivalent assumption would be that, if is any inductive class, there are objects which are not members of . For in that case, if be such an object, . Hence by induction, every inductive cardinal must exist. Another equivalent assumption would be that (the class of all objects of the type in question) is not an inductive class. The assumption that exists in the type in question is, as we shall see, a stronger assumption than the above, unless we assume the multiplicative axiom.

If the axiom of infinity is true, the inductive cardinals are all different one from another, i.e. , where and are inductive cardinals, is not equal to unless . But if the axiom of infinity is false, then, in any assigned type, all the cardinals after a certain one are . (Except in the lowest type, the last existent cardinal must be a power of 2.) That is, if (say) 8 were the largest existent cardinal in the type in question, we should have, in that type, , and the same would hold of 10, 11, .... This possibility has to be taken account of in what follows.

In order to give typical definiteness to the axiom of infinity, we write

*120·04.

Then "" states that, if is any inductive cardinal, there are at least objects of the same type as .

*120·1.

*120·101.

The right-hand side of the above equivalence gives the usual formula for mathematical induction. Observe that the conditions of significance require that should be taken in the same type as . This fact is specially relevant in the proof of *120·15.

The symbol "" is of ambiguous type not necessarily the same in different occurrences; also, according to the convention explained in the prefatory statement as holding for and , "" will not imply that and are of the same type. Accordingly to avoid error in connection with *120·1·101 typical definiteness is required as in the three following propositions.

*120·102.

*120·103.

*120·11.

*120·12.

*120·121.

By means of this proposition and *120·12, any assigned cardinal in the series of natural numbers can be shown to be an inductive cardinal; thus e.g. to show that 27 is an inductive cardinal, we shall only have to use *120·121 twenty-seven times in succession.

*120·122.

*120·123.

*120·124.

Dem.

*120·13.

Dem.

The above proposition is often convenient for inductive proofs.

*120·14.

Dem.

This proposition does not show that every inductive cardinal is an existent cardinal; to obtain this, we require the axiom of infinity.

*120·15.

i.e. a cardinal which is not null and is inductive in any one type is also inductive in any other type.

Dem.

*120·151.

Dem.

*120·152.

Dem.

The following propositions, giving alternative forms for the definition of inductive classes, are inserted in order to show that the theory of inductive classes might be treated in a less arithmetical manner than we have adopted.

*120·2.

*120·201.

Dem.

*120·21.

Dem.

Note that "" is not proved above. The proof encounters the difficulty that we may have ; in order to establish our proposition in this case, we have to show that if , then every class is an inductive class. We can however prove the following implication.

*120·211.

Dem.

*120·212.

*120·213.

*120·214.

The following propositions are lemmas for *120·24.

*120·22.

Dem.

*120·221.

Dem.

*120·222.

Dem.

The proof of this proposition might also proceed by the use of uniform formal numbers, employing *118·241.

*120·23.

Dem.

*120·24.

Dem.

This proposition might be used to define inductive classes. It gives a form of mathematical induction applicable to classes instead of to numbers. Virtually it states that an inductive class is one which can be formed by adding members one at a time, starting from . This is made more explicit in *120·25. Instead of , in the above propositions, as well as in those that follow, we may plainly substitute

*120·25.

*120·251.

*120*26.

*120·261.

*120·27.

Dem.

This proposition also follows immediately from *12·21·15.

*120·3.

*120·301.

*120·31.

Dem.

*120·311.

*120·32.

Dem.

*120·321.

Dem.

*120·322.

*120·33.

*120·41.

Dem.

The above proposition establishes (with the natural limitations) the uniqueness (within each type) of subtraction (conceived as in *120·412) when the subtrahend is an inductive cardinal. (When the subtrahend is a non-inductive cardinal, subtraction ceases to give a unique result.) Hence we are led to the following extensions of *118 for the case of inductive cardinals:

*120·411.

Dem.

*120·4111.

Dem.

*120·412.

Dem.

*120·413.

Dem.

*120·414.

*120·415.

*120·416.

*120·417.

*120·418.

*120·42.

Dem.

*120·422.

Dem.

*120·423.

Dem.

*120·4231.

Dem.

*120·4232.

*120·424.

Dem.

*120·425.

Dem.

*120·426.

Dem.

*120·427.

The above proposition shows that no reflexive class is inductive.

*120·428.

Dem.

*120·429.

Dem.

The following definition, in which "" stands for "," defines the "species" of a cardinal as all cardinals which are less than, equal to, or greater than . We cannot prove, unless by assuming the multiplicative axiom, that all cardinals belong to the species of , except in the case where is an inductive cardinal. In all other cases there may, so far as is known at present, be other cardinals which are neither greater nor less than .

*120·43.

*120·431.

*120·432.

*120·433.

*120·434.

*120·435.

*120·436.

*120·437.

*120·438.

Dem.

*120·44.

Dem.

*120·441.

*120·442.

Dem.

*120·45.

Dem.

*120·4501.

Dem.

The following proposition is a lemma in the proof of *120·452.

*120·451.

Dem.

This proposition could be extended to greater generality as regards types; but its sole use is as a lemma.

*120·452.

Dem.

In the last line but one of the above proof, we substitute for the of *120·11 the function

The following propositions are chiefly required as leading to *120·4621·4622·47, which are useful in proving propositions concerning all inductive cardinals other than zero.

*120·46.

Dem.

*120·461.

Dem.

*120·462.

*120·4621.

Dem.

*120·4622.

Dem.

It is on this proposition that the irrelevance of types in the consideration of inductive cardinals depends.

*120·463.

*120·47.

Thus mathematical induction starting from 1 will apply to all inductive cardinals except 0. Similar propositions can be similarly proved for 2, 3, ....

*120·471.

Dem.

*120·472.

Dem.

*120·473.

Dem.

*120·48.

Thus every cardinal which is not greater than every inductive cardinal is an inductive cardinal.

*120·481.

Thus if any inductive class can be found which contains a given class, the given class is also inductive.

*120·49.

Dem.

Thus every non-inductive cardinal (except ) is greater than every inductive cardinal (except ).

*120·491.

Dem.

*120·492.

In virtue of *120·491, a class which is not inductive contains sub-classes having 0, 1, 2, 3, ... terms. If we take the successive classes of sub-classes these are mutually exclusive, and all exist provided is not an inductive cardinal, i.e. provided the axiom of infinity holds. Thus if the axiom of infinity holds, we get classes of sub-classes contained in any non-inductive class. It follows, as we shall see later, that if is a non-inductive class, is a reflexive class. This seems to be the nearest approach possible to identifying the two definitions of finite and infinite when the multiplicative axiom is not assumed. When the multiplicative axiom is assumed as well as the axiom of infinity, we pick out one class from , one from , and so on; then, forming the logical sum of all these classes, we get terms which are members of . Hence it follows that is a reflexive class; for, as we shall see later, a reflexive class is one which contains sub-classes of terms. Thus with the help of the multiplicative axiom, the two definitions of finite and infinite can be identified.

*120·493.

Dem.

*120·5.

Dem.

The restriction involved in in the hypothesis of the above proposition is not necessary if we assume that the axiom of infinity must fail in any one type if it fails in any other, i.e. where and are any two objects of any two types. To prove this proposition would require assumptions, as to the interrelation of various types, which have not been made in our previous proofs.

*120·51.

This proposition establishes the uniqueness of division among inductive cardinals.

Dem.

If , in the above are typically ambiguous symbols, such as we have ; for in this case, . Also if and are of the same type, we have , in virtue of *103·43. Hence "" may, with truth, be substituted for "" in the above proposition, since the result is true whenever significant. But in this form the proposition gives less information, since it tells us nothing as to what happens when and are not of the same type.

*120·511.

Dem.

*120·512.

Dem.

*120·513.

This proposition does not hold when is a non-inductive cardinal.

*120·52.

Dem.

*120·53.

Dem.

If , , are typically ambiguous symbols, we have in the conclusion of the above proposition, instead of . Also if and are of the same type, ; thus whenever "" is significant.

*120·54.

For the proof, which is here given shortly, compare *117·58.

Dem.

*120·541.

*120·542.

*120·55.

Dem.

*120·56.

Dem.

*120·561.

Dem.

*120·57.

Here "" is necessarily in a higher type than "," because it applies to a class of which is a member.

Dem.

*120·6.

Dem.

*120·61.

Dem.

*120·611.

Dem.

*120·62.

Dem.

The above proposition may also be deduced from *120·611, by *62·231.

*120·63.

In virtue of this proposition the multiplicative axiom is not required in dealing with a finite number of factors, even when some or all of the factors are themselves infinite.

*120·64.

Dem.

Thus of our two arithmetical axioms, the multiplicative axiom and the axiom of infinity, at least one must be true.

*120·7.

*120·71.

Dem.

The above proposition is frequently used.

*120·72.

Dem.

*120·721.

Dem.

*120·73.

*120·731.

*120·74.

Dem.

*120·741.

Dem.

*120·75.

Dem.

*120·76.

Dem.

The following propositions are concerned in establishing the converse of *120·76 subject to a suitable hypothesis. The final outcome is given in *120·77.

*120·761.

Dem.

*120·762.

Dem.

*120·764.

*120·765.

*120·766.

Dem.

*120·767.

*120·77.