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

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?

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I assume he doesn't give any examples, thats poor practise on Russels part, if he's talking about such concrete entities as the integers. – Mozibur Ullah Dec 6 '12 at 21:48