[FOM] Negative Types

Thomas Forster T.Forster at dpmms.cam.ac.uk
Mon Sep 23 04:11:05 EDT 2002

The theory of negative types is consistent by compactness
(Wang, MIND ~1952).  However all its models are nonstandard
in the sense that Type n+1 cannot reliably be the same size
as the power set of Type n.   The proof relies on Sierpinski-
Hartogs and does not depend on the axiom of foundation.  
   I published this in ZML v 35 (1989) pp 385-6.  It seemed
to me at the time to be a novel cute factoid, and worth
publishing.  If anyone knows of an earlier prrof i'd like
to know.   Tonny Hurkens has subsequently given a smoother
proof of the remark from which the nonstandardness result
follows, namely that the relation ``there is an embedding
from the power set of x into y" is wellfounded.  This is
in the Jahrbuch der Kurt Goedel Gesellschaft of a couple of
years ago.  I prefer his proof to mine.....

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