Volume 1 / Part 3 — *117. GREATER AND LESS.
Reading edition
*117. GREATER AND LESS.
Summary of *117.
A cardinal is said to be greater than another cardinal when there is a class which has terms and has a part which has terms, while there is no class which has terms and has a part which has terms. The relation "greater than" is transitive and asymmetrical; and by the Schröder-Bernstein theorem, if is greater than or equal to , and is greater than or equal to , then . But we cannot prove that of any two cardinals one must be the greater, unless we assume the multiplicative axiom. The proof then follows from Zermelo's theorem that on that assumption every class can be well-ordered. This subject will be dealt with at a later stage.
The form of the definitions is so arranged as to allow of the inequality of two cardinals in different types. The relevant considerations are the same as for the definitions of addition, multiplication and exponentiation.
Our definition of "" is
*117·01.
We also define "" as meaning "," and "" as meaning "," for the reasons explained in *110. It then easily follows that if , and must be homogeneous cardinals (this is part of *117·15); that if and are homogeneous cardinals, and , the same holds if we substitute and for one or both of and (*117·16); that
*117·13.
and that
*117·14.
We cannot define "" as "," because "" restricts and too much by requiring that they should be of the same type, and restricts them too little by not requiring that they should both be existent cardinals. To avoid both these inconveniences, we put
*117·05.
The use of this definition is chiefly through the propositions
*117·108.
*117·24.
In *117·2, we repeat the Schröder-Bernstein theorem (*73·88), which is required in most of the remaining propositions of this number. It leads at once to the propositions
*117·22.
(which practically supersedes the definition of "")
*117·221.
*117·222.
*117·23.
This last proposition may be called the Schröder-Bernstein theorem with as much propriety as *73·88; the two are scarcely different.
If we now revert to the definition of , or to *117·13, and apply *117·22, we see (*117·26) that "" may be conveniently regarded as asserting ; in fact, the best ideas to work with are and its converse , which for practical purposes we regard as defined by *117·22, and from which we derive and . The relation will be the product of into the negation of its converse; this holds for and (*117·281) as well as for and .
*117·3·31 constitute an important use of *110·72, namely to prove that one existent cardinal is greater than another or equal to it when the first can be obtained by adding to the second (where what is added must be a cardinal). That is to say, we have
*117·3.
*117·31.
*117·4—·471 are concerned in proving that and are transitive, that is asymmetrical (*117·42), and allied propositions.
Our next set of propositions is concerned with 0 and 1 and 2. We prove that a homogeneous cardinal is whatever is greater than or equal to 0 (*117·501); that a homogeneous cardinal other than 0 is whatever is greater than 0 (*117·511); that a homogeneous cardinal other than 0 is whatever is greater than or equal to 1 (*117·531); and that a homogeneous cardinal other than 0 and 1 is whatever is greater than 1 (*117·55), and is whatever is greater than or equal to 2 (*117·551).
We next prove a set of propositions concerning which have no analogues for , except when the cardinals concerned are finite. Thus e.g. we prove
*117·561.
If we substitute for , this no longer holds. Thus e.g. put , , (cf. *123); then , but . Similar remarks apply to the analogous propositions (*117·571·581·591) on multiplication and exponentiation.
We prove next that a sum is greater than or equal to either of its summands (*117·6); that a product neither of whose factors vanishes is greater than or equal to either of its factors (*117·62); that, assuming and are existent cardinals, then if they are neither 0 nor 1, their product is greater than or equal to their sum (*117·631), and if is neither 0 nor 1, then (*117·652).
The last important proposition in this number is Cantor's theorem
*117·661.
which follows immediately from *102·72 and *116·72.
The propositions of this number are much used in the following section, on finite and infinite.
*117·01.
*117·02.
*117·03.
*117·04.
*117·05.
*117·06.
The analogues of *117·02·03 are to be applied also to *117·04·05·06.
*117·1.
*117·101.
*117·102.
*117·103.
*117·104.
*117·105.
*117·106.
*117·107.
Dem.
*117·108.
*117·11.
Dem.
*117·12.
Dem.
*117·121.
Dem.
The above proof is given shortly because it proceeds on the same lines as *117·12. In applying *10·55, the of that proposition is replaced by , and the is replaced by
*117·13.
Dem.
*117·14.
*117·15.
Dem.
The advantage of this proposition is that it expresses "" in terms of and alone, without the auxiliary and of the definition.
*117·16.
*117·2.
This proposition (which is the Schröder-Bernstein theorem) is fundamental in the theory of greater and less.
*117·21.
*117·211.
Dem.
*117·22.
Dem.
*117·221.
*117·222.
*117·23.
*117·24.
Dem.
*117·241.
*117·242.
*117·243.
*117·244.
*117·25.
Dem.
*117·26.
Dem.
*117·27.
*117·28.
*117·2.
*117·29.
*117·291.
*117·3.
Dem.
*117·31.
Dem.
*117·32.
Dem.
The above proposition shows that if a cardinal exists in a given type, so do all smaller cardinals.
*117·4.
Dem.
*117·41.
*117·42.
Dem.
*117·43.
*117·44.
*117·45.
Dem.
*117·46.
*117·47.
*117·471.
*117·5.
Dem.
*117·501.
*117·51.
Dem.
*117·511.
*117·52.
Dem.
*117·53.
Dem.
*117·531.
Dem.
*117·54.
Dem.
*117·55.
Dem.
*117·551.
Dem.
*117·56.
Dem.
*117·561.
The proof of *117·561 follows from *117·56 in the same way as the proof of *117·31 follows from *117·3. In the remainder of this number we shall omit proofs of this kind.
*117·57.
Dem.
*117·571.
*117·58.
Dem.
*117·581.
The two following propositions are lemmas for *117·59.
*117·582.
Dem.
*117·583.
Dem.
*117·59.
Dem.
The hypothesis is essential in the above proposition, for while , so that .
*117·591.
*117·592.
Dem.
The above proposition is used in *120·53.
*117·6.
Dem.
*117·61.
*117·62.
Dem.
*117·63.
Dem.
*117·631.
The two following propositions are lemmas for *117·64.
*117·632.
Dem.
*117·633.
Dem.
*117·64.
Dem.
*117·651.
Dem.
*117·652.
*117·66.
Dem.
*117·661.
The above proposition is important.
*117·67.
Dem.
*117·68.
Dem.
*117·681.
*117·682.
Dem.
*117·683.
*117·684.
The above proposition is used in *120·765.