What does Russell mean when he define the “Posterity… with respect to the immediate predecessor”?

Question

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:

1. Is the posterity notion different from the notion of successor?. For me, Posterity means all successors and successors of successor of a given number.

2. What does he mean by "respect to the relation immediate predecessor"?, Why not just: "the posterity of a given number as..."

3. 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?

• 5 belongs to the hereditary class that contains 0, e.g. p0={0,1,2,3,4,5,...}
• 5 belongs to the hereditary class that contains 1, e.g p1={1,2,3,4,5,.....}, p0
• 5 belongs to the hereditary class that contains 2, e.g p2={2,3,4,5,.......}, p1, p0
• ...
• 5 belongs to the hereditary class that contains 5, e.g p5={5,.......}, p4,p3,p2,p1,p0
• 5 belongs to the hereditary class that contains 6? only 6 class satisfies = p0, p1, p2, - p3, p4, p5. Since "6" is member of p0, p1, ...p5 and "5" is member of p0, p1, ...p5.

Then, the posterity of "5" are the members of p0, p1, ... p5. In other words: "the natural numbers" but according to Russell the posterity of 5 is 5 and all numbers greater than 5.

It must be that I am not understanding what Russell wants to mean. Any help, please?

Answer 1

1. It is different in the sense that posterity is a set of members related by the notion of 'successor' to another member. Or in other words the notion of successor is used in the definition of posterity, but they are not the same thing.

2. Russell defines hereditary as meaning the set where if x is a member then x + 1 is a member, that is its immediate successor. With relations, if S means 'immediate successor' then xSy means x is the immediate successor of y, i.e. informally x = y + 1. So the hereditary set with relation to S would mean for any x that is a member, then y such that xSy is a member, that is that if 5 is a member then 4 is a member. This is going the wrong way for what is required in the posterity set, the idea is to have the posterity set of x contain x and all it's successors, which is achieved by using the relation 'immediate predecessor' instead.

3. 0 is not a member of every hereditary class containing 5, as you have shown in your example sets, all the numbers below 5 are not in every hereditary set containing 5, only 5 and the numbers greater than 5 are in all of these sets.

The word 'every' being the part of the definition of posterity I think you overlooked.