[FOM] automorphisms of the hyperreals?

Ben Crowell fomcrowell06 at lightandmatter.com
Mon Feb 6 09:42:43 EST 2006

Can anyone point me to any information on automorphisms of the hyperreals?
Going back on this list's archives to 1997, there was some interesting discussion
about the possible pedagogical advantages or disadvantages of using the hyperreals
in teaching calculus, and about the impossibility of providing explicit examples of
hyperreals. (This was before Kanovei and Shelah's 2003 paper.) Jon Barwise pointed
that a similar issue occurs in the complex numbers, where i and -i can't
be distinguished, so the system of complex numbers can only be defined
up to an automorphism. However, the situation with the hyperreals seems
much more complicated, both because it's not immediately obvious how
many properties the interesting automorphisms should preserve, and because
given a definition of an "interesting" automorphism group, it's not obvious how
to characterize the group structure or analytic properties, or even to prove
whether such automorphisms exist.

I spent some time trying to puzzle out the properties of automorphisms f
with these properties:
  1) for x in R, f(x)=x
  2) f preserves addition, subtraction, multiplication, and division
  3) f is continuous
Starting from there, I was able to prove a few straightforward properties
of such an f: it's external, monotonically increasing, and nowhere
differentiable. However, I was not able to prove existence, although it
seems intuitively reasonable that we could simply pick one infinite
number H, and scale it up or down by a constant. After all, in the
axiomatic approach, one simply assumes the existence of one such H. But
the whole thing gets complicated because if the idea was to scale up H
to kH, where k is real, we would still have numbers like H^H, which can't
be connected to H through any finite sequence of additions, subtractions,
multiplications, and divisions. So it seems as though we would have many
automorphisms that would act identically to one another on numbers like H
or H^2, but that would differ from each other in their actions on numbers like

Another interesting question, it seems to me, is whether such an automorphism
can have non-real fixed points (which would mean that it was of a different
flavor than the H->kH type). If so, then it seems that its structure on
the infinite and infinitesimal parts of the hyperreal line would contain
infinitely many copies of the corresponding structure on the finite part
of the line.

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