[FOM] analysis with hyperreals vs. surreals?

Ben Crowell fomcrowell06 at lightandmatter.com
Mon Jan 23 10:52:56 EST 2006

I'm studying Conway's On Numbers and Games, and he has some brief but
intriguing remarks about doing analysis with the surreals, and about
how the surreals compare with the hyperreals. I found it interesting
that in the surreals, there are infinities that are obviously
analogous to certain ordinals, whereas in the hyperreals, there is no
way to assign any absolute size to any infinities. It wasn't
immediately obvious to me whether this was an accident of the
construction of the two systems, or whether there was a deeper reason
why you couldn't have a system both (a) rich enough to do all of
analysis, and (b) rich enough to describe all the ordinals. On
further thought, it seems to me that if a system was rich enough to
do both, then it would have to be rich enough to state the continuum
hypothesis and prove or disprove it. After all, to do analysis we
need to know all the details of the structure of the number line, and
that seems incompatible with the undecidability of the continuum
hypothesis. The surreals' lack of an obvious generalization of the
notion of exponentiation seems to prevent one from using them to make
the kinds of statements that would decide the continuum hypothesis,
whereas in the hyperreals one can't make those statements because the
infinities don't have concrete sizes. Do these intuitive ideas make
any sense?

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