The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every hereditary class to which the given number belongs.
Two questions:
Is the posterity notion different from the notion of successor?. For me, Posterity means all successors and successors of successor of a given number.
What does he mean by "respect to the relation immediate predecessor"?, Why not just: "the posterity of a given number as..."
When he says: "as all those terms that belong to every hereditary class to which the given number belongs", I understand it as follow:
For example, given number 5, what is its posterity?. I will do that in two stages.
- Find the hereditary classes to which "5" belongs.
- Find those terms that belong to the classes above.
Let's do that:
Find the hereditary classes to which "5" belongs.
- $p_0 = \{0,1,2..\}$ is a hereditary class that contains 0
- $p_1 = \{1,2..\}$ is a hereditary class that contains 1
- ...
- $p_5 = \{5,6..\}$ is a hereditary class that contains 5
- $p_6 = \{6,7..\}$ is a hereditary class that contains 6
- ...
Only 6 of these hereditary classes contains 5: $p_0, p_1, p_2, p_3, p_4, p_5$
Find those terms that belong to the classes above:
- It is easy to see that those terms are: $\{0,1,2,3,....\}$
Again. I understand the notion of posterity. I know that the posterity of 5 must be $\{5,6,7...\}$. My problem is that I cannot obtain such class by following the membership function "all those terms that belongs to everey hereditary class to which the given number belongs". What is my mistake?