Volume 1 / Part 2 — *90. ON THE ANCESTRAL RELATION.
Reading edition
*90. ON THE ANCESTRAL RELATION.
Summary of *90.
If is any relation, "" is to mean " is an ancestor of with respect to ," where a term counts as its own ancestor provided it belongs to the field of . The definition of is as follows:
90·01.
That is, is to hold when belongs to the field of , and belongs to every hereditary class to which belongs; a hereditary class being a class such that , i.e. such that all successors of 's are 's.
*90·02.
This definition serves merely to decide the ambiguity between and , either of which might be meant of . It will be shown, however, that the two are equal (*90·132).
The most important propositions of this number are the following:
90·112.
I.e. if and if is a hereditary property belonging to , then it belongs to .
*90·12.
I.e. is reflexive throughout the field of , but not elsewhere.
*90·14
*90·15.
*90·151.
*90·16.
*90·163.
I.e. is a hereditary class.
*90·17.
*90·21.
*90·22.
I.e. the classes that are hereditary with respect to are the same as those that are hereditary with respect to .
*90·31.
*90·32.
*90·33.
*90·4.
*90·01.
*90·02.
*90·1.
*90·101.
Dem.
*90·102 is a lemma for *90·11.
*90·102.
Dem.
*90·11.
*90·111.
*90·112.
Dem.
*90·12.
Dem.
*90·13.
Dem.
The following proposition is a lemma for *90·132.
*90·131.
Dem.
*90·132.
Dem.
In accordance with our general convention as regards suffixes, and with the definition *90·02, means , not (.
*90·14.
Dem.
*90·141.
*90·15.
Dem.
Note that may be conveniently regarded as the 0th power of . By *50·64·65, when multiplied by it gives ; also it is contained in , , etc. I has properties, as regards relational multiplication, analogous to those of 1 in ordinary multiplication; thus to regard as the 0th power of is analogous to regarding 1 as the 0th power of , where is a number.
*90·151.
Dem.
*90·16.
Dem.
*90·161
Dem.
*90·162
*90·163
This proposition is important, since it proves that is a hereditary class.
*90·164
This proposition shows that is a hereditary class.
*90·17
Note that means (.
Dem.
*90·171
*90·172
Dem.
*90·18
Dem.
*90·21.
Dem.
*90·22.
Dem.
*90·23.
*90·23 is useful in the theory of sections of a series (*211). A section of the series generated by is defined as a class satisfying
*90·24.
Dem.
This proposition shows that if is a hereditary class which contains , then contains all the descendants of 's.
*90·25.
Dem.
*90·26.
Dem.
*90·27.
Dem.
*90·31.
Dem.
In the last line of the above proof, the process is as follows. Writing for , (2) becomes , while (3) becomes . Hence, by (2) and (3), Hence, by *90·112, , which is the proposition to be proved.
*90·311.
Dem.
*90·32.
Dem.
*90·33.
Dem.
*90·331.
*90·34.
*90·341.
*90·35.
Dem.
*90·351.
Dem.
*90·36.
*90·4.
Dem.
*90·41.
Dem.
*90·42.
Dem.