FOM: Conway's foundational ideas
Stephen G Simpson
simpson at math.psu.edu
Fri May 21 14:41:42 EDT 1999
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
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