[FOM] NBG, Subsets and Cantor's Theorem
Andrej Bauer
Andrej.Bauer at fmf.uni-lj.si
Mon Apr 9 16:11:18 EDT 2007
There is a proof following an idea of Lawvere of Cantor's theorem which
only uses bounded separation to first establish a general fact about set
theory, namely that sets form a cartesian closed category. The other bit
of knowledge that is needed is that the powerset P(A) is isomorphic to
the exponential Omega^A, where Omega is the set of truth values. Details
of the proof are explained at
http://math.andrej.com/2007/04/08/on-a-proof-of-cantors-theorem/
for those who are interested. Thus, if one axiomatizes set theory so
that the cartesian closed structure is explicit (as is usually done in
e.g. topos theory) rather than derived from powersets via bounded
separation, then Cantor's theorem becomes a simple lambda-calculus
calculation. I would therefore say that the axiom of separation is not
an essential ingridient of Cantor's theorem.
Best regards,
Andrej
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