FOM: sharp boundaries/tameness

[Harvey Friedman]

Continuation from 8:13PM 2/13/02.

Looks like I can show the following, which goes a long way to recover what
I took back.

THEOREM 1. There is no formula phi(x) of set theory such that ZFC proves
i) there is a unique x such that phi(x);
ii) the unique x such that phi(x) is a proper elementary extension of the
real field with Z (i.e., with a predicate symbol for the integers, of
cardinality the continuum).

The idea is to start with a countable transitive model of ZFC + V = L and
add omega_1 Cohen subsets of omega. Then show in this model that any
ordinally definable set of nonprincipal ultrafilters on all subsets of
omega must have cardinality >= omega_2.

Then show that from a proper elementary extension of the real field with Z
of cardinality the continuum one defines a set of nonprincipal ultrafilters
on all subsets of omega of cardinality the continuum.

In fact, this line of argument proves

THEOREM 2. It is consistent with ZFC that there is no ordinally definable
proper elementary extension of the real field with Z of cardinality the
continuum.

By an additional argument, we can work with proper extensions of the real
field with Z that are elementarily equivalent and of the same cardinality.
Such an extension defines a set of nonatomic probability measures on all
subsets of omega of cardinality the continuum.

It is also true that in the above model, any ordinally definable set of
nonatomic probability measures on all subsets of omega must have
cardinality >= omega_2. So we have

THEOREM 3. There is no formula phi(x) of set theory such that ZFC proves
i) there is a unique x such that phi(x);
ii) the unique x such that phi(x) is a proper extension of the real field
with Z that is elementarily equivalent and of the same cardinality.

THEOREM 4. It is consistent with ZFC that there is no ordinally definable
proper extension of the real field with Z that is elementarily equivalent
and of cardinality the continuum.

In fact, we can weaken "elementarily equivalent" considerably to just
satisfying the field axioms, Z is a discrete subring, and the least upper
bound principle for all formulas in the extended language.

One can also work with higher cardinalities, and, in a sense, any
cardinality, but it is not clear how to do this wihtout any cardinality
restriction.