What does Russell mean when he defines 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:


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*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.


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*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.


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*$p_0 = \{0,1,2..\}$ 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?
 A: Think of your posterity (in the ordinary sense). Who are they? Your children, and your children's children, and  your children's children's children, etc. etc. -- and no one else outside that chain (except it is convenient to count you as part of your own posterity ... and why not, since you start everything off! :-)
So take the relation $xRy$ = $x$ is a parent of $y$. Then $x$'s posterity (with respect to the relation $R$) are all the people $y$ such that either $x = y$ or $xRy$ or there is a $z$ such that $xRz \land zRy$ or there is a $z$ and $z'$ such that $xRz \land zRz' \land z'Ry$ or ... (and no one else).
Now take the numerical relation $xPy$ = $x$ is immediate predecessor of $y$ (i.e. $y$ is the successor of $x$). Then $0$'s posterity (with respect to the relation $P$, the predecessor relation), are all the numbers  $y$ such that either $x = y$ or $xPy$ or there is a $z$ such that $xPz \land zPy$ or there is a $z$ and $z'$ such that $xPz \land zPz' \land z'Py$ or .... So $0$'s posterity comprises $0$ itself and all the numbers you can get to by going from one number to the next, starting from $0$ (and comprises nothing else). In other words, $0$'s posterity are the natural numbers. 
But of course 5's posterity are 5 and its successor-descendants (which don't include $0$).
OK: that's the intuitive notion of posterity (or transitive closure under a relation). So far so good?
Now, let's ask: can we give a tidier official definition, without having to arm-wavingly trail off using '...' because we can't write down an unending series of conditions?
Yes. Say a class $C$ is hereditary with respect to a relation $R$ if, whenever it contains $x$ and $xRy$ it contains $y$. Then (think about it) your posterity is the smallest class which contains you and which is hereditary with respect to the relation $xRy$ = $x$ is a parent of $y$. Likewise (think about it) the natural numbers form the smallest class which contains $0$ and which is hereditary with respect to the relation $xPy$ = $x$ is the immediate predecessor of $y$.
Finally, what is it to say that the natural numbers form the smallest $P$-hereditary class containing $0$. It is of course to say that a natural number belongs to every $P$-hereditary class containing $0$. Which is what Russell (very clearly!) says, closely following Frege at this point.
A: But $0$ does not belong to the posterity of 5, since the set $A=\{5,6,7,...\}$ is a particular hereditary set containing $5$ and $A$ does not contain the element $0$. The key phrase is all those which "belong to every hereditary set containing the given number", and we have this particular set $A$ which goes under the term every here, yet it doesn't contain $0$.
I agree it seems pecular here for Russell to say "posterity with respect to immediate predecessor", and it seems to me it would be more apt to say "posterity with respect to the relation successor", or as you say, simply use the term "posterity" of a number for this concept.
Maybe the "posterity" relation can be viewed as the transitive closure of the original relation $xRy$ defined to mean $y$ is the successor of $x$. I've seen the term transitive closure more often than the term "posterity", and maybe Russell was coining his own term, either because he was one of the early writers on set theory, or because he wanted to use a nontechnical term in a book on the introduction to philosophy. 
