No subject
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Fri Mar 1 10:15:59 EST 2002
I've been thinking about Specker's 1959 paper `Dualitaet'
recently - I translated it and wrote a commentary for the Taylor
and Francis four volume Quine bash and they have let me put
a .ps file on my home page: dpmms.cam.ac.uk/~tf/dualitaet.ps.
One of the points Specker makes is that although if a theory $T$ has
an automorphism $\sigma$ then for any fmla $\Psi$ whatever, T proves
$\Psi$ iff T proves $\sigma(\Psi)$, this is certainly not enough to
ensure that T proves $\Psi \iff \sigma(\Psi)$. Special conditions
apply...
Specker considers duality in geometry but the issues arise in other
connections. Clearly any recursive permutation of the natural numbers
gives rise to an automorphism of any theory of syntax that has been
coded up as a theory of arithmetic. Change the Goedel numbering by
composing with a recursive permutation and get a different Goedel
sentence. One is deducible iff the other is (and of course neither of
them is!) but is this situation special enough to ensure that the
biconditional between them is provable? Presumably the answer is
known (tho' not to me!) And is the situation didfferent for Rosser
sentences?
Thomas
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