FOM: why Con(ZFC) isn't finite combinatorics

Stephen G Simpson simpson at
Tue Aug 18 17:04:19 EDT 1998

Joseph Shoenfield writes:

 > ConZFC is a statement about the arrangement of objects called
 > sentences of the language of set theory into patterns called proofs
 > in ZFC.  Steve has nothing to say about why ConZFC does not fit his
 > definition of a finite combinatorial statement.

Hmmmm.  I guess I thought this was obvious, but maybe it's difficult
to explain.  Let me take a quick stab at it now, and maybe I'll make
another attempt later.

The reason Con(ZFC) isn't a finite combinatorial statement is that it
can't be understood or appreciated in terms of the nexus of problems
and concepts which form the subject of finite combinatorics as
commonly understood by mathematicians, e.g. in the work of
combinatorists such as Graham, Lovasz, Spencer, ..., i.e. concepts and
problems concerning finite graphs, finite arrangements of objects into
patterns, etc.  The only way to grasp Con(ZFC) is in terms of a
picture of the universe of Zermelo/Fraenkel set theory.

Sorry, that's the best I can do right now.

-- Steve

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