[FOM] paradox and circularity

[Karl Cooper]

Steven Yablo writes:

> A set S of  type k is well-founded if there are no sets S_(k-1), 
> S_(k-2), etc. such that S contains S_(k-1) contains S_(k-2) and so on 
> forever.
> 
> 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.
> 
> 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.  So no G_n is well-founded.  Contradiction.
> 
I reply:

Consider the case in which the empty set is defined (or declared
by an axiom) to be the only set of type zero.

Then there are no sets of type -1, -2, etc.

But there are certainly sets of type 1, 2, etc.

And no set of type n need lead to an infinite chain of sets of
ever-decreasing type, since all such chains stop at zero.

Karl Cooper
Senior Software Engineer
interests in set theory