[FOM] paradox and circularity
KCooper at transcept.com
Mon Sep 16 08:52:30 EDT 2002
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
> 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.
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.
Senior Software Engineer
interests in set theory
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