[FOM] automorphisms of hyperreals [FROM ALI ENAYAT]
a_enayat at hotmail.com
Sat Feb 25 23:01:29 EST 2006
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
(Q1): Do the nonstandard reals admit a nontrivial automorphism?
(Q2): Can a nontrivial automorphism of nonstandard reals have a nonstandard
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
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
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