FOM: sharp boundaries/tameness

Kanovei kanovei at
Thu Feb 14 15:27:16 EST 2002

>From: Harvey Friedman <friedman at>

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.

This seems true and should be doable but most usual tricks 
which prove such results for sets of reals do not apply 
because while "old" reals still are reals in the extension, 
"old" ultrafilters are not at all ultrafilters in the extension. 

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

Taking T1 on trust, the extension even needsn't to be proper. 

The rest of the scheme looks too sketchy. 
Let, by analogy, Friedman's set of a nonstandard model of analysis 
be the set of all ultrafilters on N obtained certain way from all 
hyperintegers of the model. 
(Or perhaps all finitely additive measures on P(N).) 

It can be reasonably expected that such a strange object as 
a definable nonstandard extension of R is Friedman-complete, 
that is, its Friedman algebra contains all ultrafilters, in 
particular, has cardinality aleph-2 in the model as above. 


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