*90. ON THE ANCESTRAL RELATION.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
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.