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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. Dec 6 '12 at 21:48
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  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.

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When plain English is used to explain mathematics, the effect is often to the opposite of what the author intended. For lucid explanation, see "Section E Inductive Relations" of *Principia Mathematica by Whitehead & Russell. Symbols are a lot easier to understand than "plain English." Yes, you can delve directly into this section. I'll be happy to explain those symbols.

The summary of this section reveals what mountains of labour were behind a simple definition.* For example, W&R discovered that the "intuitive" conception of an ancestral relation conceals the concept of finitude, whose very definition depends on the ancestral relation itself. Thus the definition of ancestral relation familiar to our common sense will not do.

  1. Yes, they are different. A successor is an individual; the posterity of an individual is a class. Take the relation father-to-son for example, Conrad Russell is a successor to Bertrand Russell but is not a successor to Lord John Russell because Lord John Russell is not the father of Conrad Russell; both Bertrand Russell and Conrad Russell are members of Lord John Russell's posterity.

    1.1 With respect to the relation "immediate predecessor," 0 is the immediate predecessor of 1; 1 is the immediate successor of 0, whereas the posterity of 0 are all the natural numbers, including 0.

  2. Given a relation R, we write xRy to denote "x has to y the relation R." we say a is an ancestor of z with respect to R if there is a certain number of intermediate terms b, c, d, ...y such that aRb, bRc, cRd, ... yRz. Different relations generate different hereditary classes over the same field. When speaking of posterity, it is necessary to point out the relation to which the posterity is with respect. For example, Theodore Roosevelt is one of George Washington's descendants with respect to the relation of successor-of POTUS, but not with respect to the relation father-to-son.

    2.1 When xRy, we say "x precedes y" or "y succeeds x" with respect to R. Notice that "succeeds" does not imply "immediate succeeds." For example, with respect to "less than," 2.5 is less than 3, but 3 does not immediately succeeds 2.5; actually 2.5 has no immediate successor with respect to "less than" over the field of rationals. 1's immediate successor is 2 with respect to the relation "-1."

    2.1.1 In virtue of ✳38.101 ⊢: u (♀y) x .≡. u = x♀y

    x (-2) y is equivalent to x = y - 2. Thus, with respect to the relation "-2", the posterity of 0 includes 2, 4, 6 but not 5, which is a descendant of 3. With respect to the relation "-5", the posterity of 0 includes 5, 10, 15, ... but not 6, which is a descendant of 1. Notice that, strictly speaking, the relation "-2" in W&R's language is defined as x̂ŷ(x=y-2); the relation of "-1" is defined as âb̂(a=b-1) or âb̂(b=a+1), thus if b is a descendant of 0, b can be reached by repeatedly adding 1 to 0.

    2.2 Plain English often gives people the impression of "going the wrong way" because xRy, in plain English, becomes "y succeeds x" with respect to R, where the typographic order is reversed.

  3. "Every" implies intersection. Students who are members of every club constitute the intersection of these clubs. 0 does not belong to p5 of which 5 is a member, thus 0 is ruled out; the same can said about 1, 2, 3 and 4.

A class μ is said to be hereditary with respect to R if any term y to which a member of μ has the relation R is also a member of μ.

A property defines the extension of a class. A class μ is hereditary with respect to relation R in virtue of the fact that the property ϕ that defines this class is hereditary with respect to R. In other words, if x is a member of μ define by ϕ, and x has the relation R to y, then ϕ(y), i.e. y is also a member of μ. If y has the relation R to z, then ϕ is also a property of z, i.e. z is a member of μ.

Being a member of a class defined by ϕ is equivalent to having the property ϕ; z belongs to every hereditary class to which a belongs is equivalent to z has all the hereditary properties possessed by a. To have a for one's ancestor is a hereditary property possessed by a, thus if z belongs to every hereditary class to which a belongs, then z has the property of having a for its ancestor—this explains what the ancestral relation entails and why the ancestral relation is defined in such a seemingly "unnatural" way.

Hereditary class is a very rich concept. It is extremely rewarding to set aside a few peaceful days (or weeks) to meditate on it.

  • I do not pretend that W&R were not standing on the shoulders of some giants. Chances are either Peano, Cantor or Frege had conceived something similar before W&R.

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