[FOM] Automorphisms of nonstandard reals, revisited
a_enayat at hotmail.com
Tue Feb 28 22:51:21 EST 2006
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
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.
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