Re: Internal Set Theory Uniqueness Principle

If you accept the Hilbert tau operator (as in Bourbaki), the answer is
obviously no. For instance, define epsilon to be that x such that x is
a non-zero infinitesimal. Let p(x) be x = epsilon. Actually, I believe
all you would need to do would be to add what I believe are called
Henkin constants (essentially the constants made from tau operators
applied to formulas with only one free variable, as is done to get
conservative extensions that are Henkin theories). But without such
constructions, and more particularly, without such constructions being
allowed to be applied to external formulas (which should lead to a
broader standardization principle if as seems reasonable,
standardization should be allowed to be used on external formulas
involving the tau operator-but perhaps it isn't totally obvious
adding tau operators for external formulas and extending the definition
of external formula in obvious way would lead to a conservative
extension), it seems doubtful you would be able to get at a definite
infinitesimal, and that your result would thus hold, though I can't
think off hand how I'd go about showing it.

.