[FOM] Higher Order Set Theory
Santiago Bazerque
sbazerque at gmail.com
Thu Mar 10 10:32:43 EST 2005
On Wed, 9 Mar 2005 16:25:41 -0500
Dmytro Taranovsky <dmytro at MIT.EDU> wrote:
> The problem is whether higher order statements have any meaning or
> are just symbols on paper.
Does anybody know if ZFC is effectively inseparable? I find this
question interesting because if it were, then if ZFC + { any
axiomatizable extension } is consistent, it is recursively isomorphic to
ZFC (i.e. they would share the same deductive structure) by a result of
Pour-El and Kripke (Fund. Math. LXI p. 142).
This is not merely an interpretation of the extension in ZFC, but a
recursive 1-1 mapping I reinterpreting sentences of our new set theory
onto sentences of ZFC that preserves valid deductions, and a bijective
(non-recursive) mapping J between the classes of models of both theories
such that for every sentence alpha and model M of ZFC+{x},
alpha |= M iff I(alpha) |= J(M).
I am of course not saying that this would imply the "symbols on paper"
thesis, but it is new to me and is making me feel a bit uneasy about the
matter.
Sincerely,
Santiago
ps. The result above would probably still hold for any reasonably
axiomatized extension of the langage of ZFC as well, by the way.
More information about the FOM
mailing list