FOM: Large and small

[JoeShipman@aol.com]

In a message dated 5/3/99 6:02:03 PM Eastern Daylight Time, fenner at cs.sc.edu 
writes:

<< 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
 each
 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