FOM: Large and small

Stephen Fenner fenner at
Mon May 3 16:32:23 EDT 1999

On Fri, 30 Apr 1999, Joe Shipman wrote:

> Simpson:
> >If you insist on talking about large
> >categories, you might as well also insist on talking about large
> >groups, large rings, large fields, etc.  [ I seem to remember somebody
> >saying that John H. Conway's surreal numbers form a large field .... ]
> Yes, and Conway (in "On Numbers and Games") casually refers to Groups,
> Fields, etc., with initial capitals when the domain is a proper Class to
> distinguish them from set-size groups, fields, etc.

If memory serves me, Conway's surreal numbers can be built in layers
H_alpha for alpha any ordinal, with

H_0 subset H_1 subset H_2 subset ...

and taking unions at limits.  The entire union H is a proper class, but
H_alpha is a set (a real closed field for limit alpha(?)).

I think a reasonable question to ask in this case is: Is there any
interesting property of H that doesn't reflect down to H_kappa for some
hereditarily accessible kappa?  If not, then the bigness of H doesn't
really amount to anything interesting.

Steve F

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