[FOM] paradox and circularity

[Todd Wilson]

On Fri, 13 Sep 2002, Stephen Yablo wrote:
> For each integer n, let G_n be the set of well-founded sets of type (n-1).
> 
> On the one hand, each G_n must be well-founded, because an infinite 
> descending membership chain starting from it would include an 
> infinite descending membership chain starting from one of its 
> members, and its members are one and all well-founded.

Yes.
 
> On the other hand, if each G_n is well-founded, then it belongs to 
> the set of well-founded sets one level up, that is G_n belongs to 
> G_(n+1).  Since n here ranges over the integers this gives us an 
> infinite descending chain: each G_k contains G_(k-1) contains G_(k-2) 
> etc.

Well, what are the sets of type -1, -2, ...?  Presumably, types start
at 0, in which case there is no infinite descending chain.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh