FOM: Large and small
JoeShipman at aol.com
Tue May 4 00:36:48 EDT 1999
In a message dated 5/3/99 6:02:03 PM Eastern Daylight Time, fenner at cs.sc.edu
<< 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.
Read Conway's book "On Numbers And Games" -- surprisingly large ordinals crop
up very quickly. The more structure Conway builds up, the larger the
surreals have to be in order to be closed under all the relevant operations.
The surreal numbers are highly set-theoretical in flavor and ordinals play a
critical role; his proper Class No of surreal numbers probably has the same
kinds of reflection properties V does, and the free use of proper Classes
very significantly eases the exposition.
One gets the impression from his passing foundational remarks that he
wouldn't think it a big problem for a theory to require one or more
inaccessibles to be properly formalized, and it is obvious that ZF plus a
couple of inaccessibles would suffice for his development, but in fact VNBG
would also work with a little care, as Conway is clearly aware.
Conway proposes that proper classes, or Universe-size sets, be referred to as
"Universities" by analogy with "infinities", so that there is an infinity of
finite ordinals but a University of ordinals (sounds sort of like a college
of cardinals--this was one of the inspirations for my definitions of
"eligible" and "infallible" cardinals a few months back). Conway calls
collections which are larger than Universe-size "Improprieties", since they
are not proper Classes, but does not attempt to do anything with the one such
collection he defines because it is "an illegal object in most set theories".
-- Joe Shipman
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