[FOM] Higher Order Set Theory

[Santiago Bazerque]

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.