FOM: {n: n notin f(n)} ADDENDUM
Richard Heck
heck at fas.harvard.edu
Tue Aug 27 17:54:57 EDT 2002
Andrew Boucher pointed out what I believe to have been a mistake in my
previous posting. I remarked there that, in the instance of separation
one needs for Cantor's argument, the formula one needs is
"quantifier-free". As Andrew notes, that does not appear to be true, at
least in the most obvious way of formalizing the argument: The formula
"n notin f(n)" hides a quantifier, because f is really an ordered pair.
Maybe there is some trick that will reduce complexity here. I don't know.
On the other hand, the following is true: If have power set, then the
quantifier in question can be bounded, and the formula in question will
be delta-0. So rejecting Cantor's argument as Dean suggests means
rejecting delta-0 instances of separation, which makes the set-theory
quite weak, though not of course completely hopeless. One could still
accept "open" instances.
So here's a very open-ended question: How much is known about such weak
fragments of set-theory? Say, ZF, with separation limited to open, or
delta-0, formulae? Any helpful references?
Richard
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