[FOM] Automorphisms of nonstandard reals, revisited

[ali enayat]

In an earlier posting [Feb 25, 2006] I noted that rigid nonstandard models 
of analysis exist, if there is a Ramsey ultrafilter (recall: Ramsey 
ultrafilters exist in ZFC + CH, and in ZFC + MA + not CH).

Recall that a nonstandard model of analysis is simply a proper elementary 
extension of (R,X), where X varies over finitary relations over R [in 
particular, there is a distinguished predicate for the set of natural 
numbers N, and for the set of rationals Q].

I now see that, one can do a lot better than I initially claimed: for *any* 
nonprincipal ultrafilter U on P(N) [the powerset of N], the ultrapower of 
the standard model of analysis by U is rigid.

Therefore rigid models of analysis exist outright in ZFC. Moreover, the 
iterated ultrapower of the standard model of analysis modulo U along a rigid 
linear order L, is itself rigid. Therefore, there are arbitrarily large 
rigid nonstandard models of analysis [recall: the order type of any aleph is 
rigid].

The main technical tool in establishing the above is a lemma, due 
independently to Ehrenfeucht and Gaifman that states: any *simple* 
elementary extension of a model of arithmetic is rigid.

Here by a model of arithmetic, I am referring to any model (M,+,., ...) in a 
language L, that satisfies PA(L), where PA(L) is PA (Peano arithmetic) 
augmented with all instances of induction formulated in L.

Recall that if (R*,N*,Q*,...) is a nonstandard model of analysis [where N* 
is the set of nonstandard "natural numbers" of R*, and Q* is the set of 
N*-rationals], then Q* is dense in R*. Hence if N* is rigid, then any 
automorphism of (R*,N*, Q*,...) fixes N* pointwise, thus fixing Q* 
pointwise,  and therefore fixing R* itself pointwise.

Regards,

Ali Enayat