The Definition of V


This account follows the definition of "V" (i.e. the collection of sets which form the iterative or cumulative heirarchy) given in lecture notes (728Kb postscript) for a course on The Foundations of Mathematics by Stephen G. Simpson.

I believe this is pretty much the standard story, except perhaps for the question of what an ordinal is. Ordinals are perhaps more often defined as certain sets (the Von Neumann ordinals) rather than as isomorphism classes.

This all hinges around two things. The first is the definition of an ordinal. The second is the power set constructor. The stages in the iterative construction are the ordinals and at each stage you get the union of the power sets of all the previous stages.

On top of any problems we may have with the ordinals and the power set constructor, there is added at the end the question whether it makes sense to suppose this process can ever be completed.

At some stage I will probably rewrite this with a different definition of the ordinals.

Relational Structure

A relational structure is a pair (A,R) where R subsetof (Acartesian productA)

Isomorphism Type

Two relational structures (A,R) and (B,S) are isomorphic if:
existsf memberof AfunctionB. (forall b memberof B. exists a memberof A. f(a)=b)
(foralla, a' memberof A.
(a = a' equivalent f(a) = f(a'))
(a R a' equivalent f(a) S f(a'))
There exists a bijection between A and B which respects (preserves) the relations R and S, i.e. maps pairs in A related under R to pairs in B related under S and vice-versa.
The isomorphism type of a relation is the class of relations which are isomorphic to it.

Well Founded

A relational structure (A,R) is well-founded if:
forallX subsetof A. existsa memberof X. not existsb memberof X. b memberof a
For every subset X of A there exists an element a which contains no element of X (a is disjoint from X).

Well Ordering

A relation (A,R) is transitive if for any three elements a,b,c in A, if a R b and b R c then a R c.
A relation (A,R) is a linear-ordering if for any two elements a,b in A eactly one of a R b, a=b and b R a holds.
A well-ordering is a linear-ordering which is well-founded.

Ordinal Number

An ordinal number is the isomorphism-type of a well-ordering.

Pure Set

A set is transitive if all its members are also subsets.
The transitive closure of a set A (TC(A)) is the smallest superset of A which is transitively closed.
A set A is well-founded if the relational structure (A, {<x,y> memberof (Acartesian productA) | x memberof y}) is well-founded.
A set is pure if every element of its transitive closure is a set.


We define corresponding to each ordinal number delta a stage V(delta) in the construction of the cumulative hierarchy as follows:
V(0) = emptyset
V(phi+1) = phi (V(phi)) for all ordinals phi
V(delta) =
phi < delta
V(phi) for limit ordinals delta
More concisely:
V(delta) =
phi < delta

The iterative or cumulative hierarchy is the union of the V(phi) for every ordinal phi.

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