[FOM] automorphisms of hyperreals [FROM ALI ENAYAT]

[ali enayat]

This a belated reply to a posting of Ben Crowell [Feb 6, 2006] who asked for 
references on automorphisms of nonstandard models of analysis.  Crowell also 
asked:

(Q1): Do the nonstandard reals admit a nontrivial automorphism?

(Q2): Can a nontrivial automorphism of nonstandard reals have a nonstandard 
fixed point?


In this discussion I take a nonstandard model of analysis to be an 
elementary extension of the structure (R,+,.,X), where X ranges over all 
finitary relations on the set of reals R.

I do not know of any papers that deal specifically with automorphisms of 
models of nonstandard  analysis, but there is a large literature on 
automorphisms of models of (nonstandard) models of Peano arithmetic PA. 
There is also a growing interest in the study of automorphisms of models of 
other strong foundational systems, such as second order arithmetic, or even 
set theory.

However, one can use a dose of classical model theory to show that 
nonstandard models of analysis with rich automorphism groups exist, and 
therefore both questions Q1 and Q2  above have an affirmative answers for 
appropriately chosen models of nonstandard analysis.

More specifically, the classical work of Ehrenfeucht and Mostowski (1956) 
shows that for every infinite model M, and any linear order L, there is an 
elementary extension M* of M such that Aut(L) is embeddable in Aut(M*) [here 
Aut(X) is the automorphism group of the structure X].

Indeed the models with rich automorphism groups can be further required to 
have any prescribed degree of saturation [by an standard compactness 
argument]. This is of interest since some of the deeper applications of 
nonstandard analysis require a degree of saturation [e.g., the Loeb measure 
construction, which needs aleph_1 saturation].

More surprisingly, assuming CH (the continuum hypothesis), or MA (Martin's 
axiom) plus not CH, one can also construct a RIGID nonstandard model of 
analysis. This is because of the fact that under CH or (MA plus not CH) 
Ramsey ultrafilters exist, and one can show that the ultrapower of the 
standard model of analysis by a Ramsey ultrafilter is rigid (note that it is 
also aleph_1 saturated).

I do not know, however, whether one can prove in ZFC alone that there is a 
rigid model of nonstandard analysis. But let me point out that it is known 
that rigid models of PA of arbitrary cardinality exist (a result of 
Gaifman).

Best regards,

Ali Enayat