FOM: Conway's foundational ideas

[Stephen G Simpson]

Joe Shipman 21 May 1999 13:20:09

 > The completeness "property" is that there are no gaps -- if the two
 > sets satisfy the ordering condition, there is a number in between
 > them 

Oh, so that's what you meant by ``completeness'' of the surreals.
Joe, thanks for the explanation.  Question: Does every bounded
nonempty set of surreal numbers have a least upper bound?  This is
what I would tend to think of as completeness.

(By the way, even though the surreal line has no set gaps, it does
have proper class gaps.  In fact Conway's book has a section entitled
``Gaps in the Number Line''.  This is where Conway goes wild with
Universities, Improprieties, etc.)

In any case, mutatis mutandis, the ordering of the hereditarily
countable surreals is ``countably complete'' a la Shipman.  In other
words, if L and R are two countable sets of hereditarily countable
surreals and all elements of L are less than all elements of R, then
there is a hereditarily countable surreal in between.

By quantifier elimination it follows that the *ordering* of the
hereditarily countable surreals is a countably saturated dense linear
ordering without endpoints.  If we assume CH, these properties
uniquely characterizes the *ordering* of the hereditarily countable
surreals, up to isomorphism.  (Proof: transfinite back-and-forth
argument of length omega_1.)

I conjecture that the hereditarily countable surreals are also
countably saturated as an ordered field.  Assuming CH, this would
likewise uniquely characterize the *ordered field* of hereditarily
countable surreals, up to isomorphism.  (Same proof.)  And many other
interesting consequences would also follow, because of the
resplendency of saturated models.  (See Chang and Keisler.)

And I think you could get the obvious analogous results for *all* the
surreals, perhaps using global choice, but with no need for GCH or any
similar assumption.

-- Steve