FOM: completeness theorem for stratification?
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Mon Apr 17 04:15:15 EDT 2000
Steve S says:
>Where does this leave us with respect to my earlier question (FOM, 7
Mar 2000) about the set-theoretic intuition? Is there a clear and
convincing picture which underlies NF, similar to how the picture of
the cumulative hierarchy underlies ZFC?
Well - as Steve evidently suspects - the completeness thm for stratified
formulae doesn't provide a picture of an NF universe the way the
cumulative hierarchy justifies the choice of axioms that make up ZF
(or at least Zermelo). What is does do is show that stratification
is not a batty idea, and leaves open the possibility that having a
set theory that says that every stratified formula has an extension
just might work.
My feeling is that the way to understand the concept of set that
underlies NF is to think through Specker's theorem that NF is
consistent iff the simple theory of types with an ambiguity scheme
is consistent. This doesn't give much of a justification, i admit,
but then i am not, in general very optimistic about justifications
for this sort of thing anyway. Just beco's something is obvious
doesn't mean it is true, and i think people who feel that the obviousness
of the cumulative hierarchy conception of set and the axioms that
flow from it are skating on ver thin ice indeed. After all, it was
obvious to my forbears that africans were fit to be enslaved, and
it`s not at all obvious to me. Geography teaches us the same lesson:
it`s obvious to Charlton Heston that humans have a god-given right
to carry guns and it's obvious to me that he's barking mad, and we
can't both be right. So: beware of obviousness, don't expect too
much of axiom systems, just be snivelingly grateful while your pet
system remains apparently consistent!
However if you are what some of my philosophical colleagues call
a ``mad dog realist" about mathematics, then you will presumably
be confident that something of which NF is true will emerge from
the mist, and your confidence in this will increase with every year
that passes in which NF remains unrefuted. Patience!
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