FOM: Large and small

Joe Shipman shipman at
Fri Apr 30 09:02:33 EDT 1999

>Here is another way to make my point.  Logically, the issue of small
>(i.e. set size) versus large (i.e. proper class size) is not really
>peculiar to category theory.  It applies to any kind of algebraic
>structure whatsoever.  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.  This never presents
a problem and the interplay between small and Large in his development
is very straightforward.  He remarks that all of his results could be
easily formalized in ZFC but doesn't regard this as necessary, and is
obviously aware of what's legal and illegal in various set theories.
Although some have interpreted his introduction, in which he talks about
"liberating" mathematicians from the constraints of formal foundational
systems, as anti-foundationalist, a careful reading of the book shows
that the work is set-theoretical to the bone, and Conway is simply
reacting against unnecessary insistence on restricting one's language so
that one has to resort to cumbersome locutions like the Kuratowski
definition of ordered pair.  (That is, it should be possible in a
foundational work like ONAG to simply INTRODUCE a concept like ordered
pair with the appropriate notation and properties, without finding an
equivalent in the language with only the epsilon relational symbol.)

I very strongly recommend "On Numbers and Games" to everyone on the FOM
list.  It is an important work in both mathematics and f.o.m., is
remarkably funny and well-written, and is one of only two math books I
always have at hand (the other is Cohen's "Set Theory and the Continuum

-- Joe Shipman

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