Volume 1 / Part 4 — *154. RELATION-NUMBERS OF ASSIGNED TYPES.
Reading edition
*154. RELATION-NUMBERS OF ASSIGNED TYPES.
Summary of *154.
This number gives propositions analogous to those of *102. In accordance with our general notations for typical definiteness, "" means "the class of relations like and of the same type as ," "" means "the relation to a relation of the type of of the class of relations like it and of the type of ." By a special definition, "" is to mean all typically definite relation-numbers of the form "," i.e. all relation-numbers generated by the relation , i.e. the domain of .
Existence-theorems in this subject can be proved by means of *154·14, which states that relations like exist in the type of when, and only when, classes similar to exist in the type of . In virtue of this proposition, the existence-theorems of our present topic are deducible from those for cardinals. In symbols, this proposition is
*154·14.
Hence by *102·73 we deduce
*154·242.
whence, by *152·72,
*154·9.
The remaining propositions are chiefly analogues of those in *102. Very few of them are subsequently referred to.
*154·01.
*154·1.
Dem.
*154·11.
Dem.
*154·12.
Dem.
*154·121.
Dem.
*154·13.
Dem.
*154·14.
In virtue of *154·14 and the propositions of *102, *103, *104, *105, *106, we see that all homogeneous or ascending relation-numbers exist, while is a member of every descending type of relation-numbers. Remembering that the relations concerned must be homogeneous, we see that there are two kinds of steps by which their types may be raised, namely (1) from to relations of the type of , i.e. from to relations of the type of , or of ; (2) from to relations of the type of , i.e. from to relations of the type of , or of if . Thus repetitions of the two steps from to , and from to , where , will enable us, without changing the relation-number, to raise its type indefinitely. It will be observed that, in accordance with our general definitions for relative types, the type of is , and the type of (where ) is .
*154·2.
*154·201.
*154·202.
*154·203.
When belongs to any other type than , is meaningless.
*154·21.
*154·22.
Dem.
*154·23.
Dem.
*154·24.
*154·241.
*154·242.
Dem.
*154·25.
*154·251.
Dem.
*154·26.
*154·261.
*154·262.
The following propositions are concerned with the two particular transformations from to and from to , which are useful in raising the type of a relation-number.
*154·31.
Dem.
*154·311.
*154·32.
Dem.
*154·321.
*154·322.
*154·33.
Dem.
*154·331.
*154·4.
Dem.
*154·401.
The remaining propositions of this number (except *154·9) are the analogues of those whose numbers have the same decimal part in *102. They are here given without proof, because the proofs are, step by step, analogous to the proofs of the corresponding propositions in *102.
*154·41.
*154·42.
*154·43.
*154·46.
*154·52.
*154·53.
*154·55.
*154·64.
*154·641.
*154·8.
*154·81.
*154·82.
*154·83.
*154·84.
*154·85.
*154·86.
*154·861.
*154·87.
*154·88.
*154·9.
Dem.