FOM: combinatorics and large cardinals

Stephen G Simpson simpson at math.psu.edu
Tue Apr 14 13:52:51 EDT 1998

```Joseph Shoenfield writes:
> I think my differences with Steve on the above topic are due to a
> different notion of what fom is about.

I think there is a more fundamental reason.

> From this point of view, finite combinatorics becomes interesting
> only when it has been precisely defined.

Here is where we part company.  I, like most mathematicians, recognize
that there is a subject called "finite combinatorics" which has a
different flavor from, e.g., elementary number theory or finite
algebra or mathematical logic.  Roughly speaking, combinatorics is the
branch of mathematics dealing with arrangements of objects in
patterns.  (Informal statements like this can be found at the
beginning of most textbooks of combinatorics).  For instance, theorems
about block designs are classified as combinatorics rather than number
theory or algebra or logic.  Lagrange's four squares theorem is
classified as number theory rather than combinatorics.  The
classification of finite simple groups is classified as algebra rather
than combinatorics.  G"odel's incompleteness theorem is classified as
mathematical logic rather than combinatorics.  These distinctions
based on subject matter are informal, not based on syntactical notions
such as Pi^0_1 or Pi^0_2, but interesting and important nonetheless.
At least I find these informal distinctions interesting and important,
even if you don't.

> ConZFC is pi-0-1, and I think it is clearly a finite combinatorial
> statement.

I disagree.  Con(ZFC) is not a finite combinatorial statement in the
informal sense that I mentioned above.  Con(ZFC) is of interest only
to people who know a fair amount of mathematical logic.  It's not of
interest to finite combinatorists qua finite combinatorists.  You can
find this out by talking to some finite combinatorists (people
interested in Ramsey theory, block designs, etc.) and trying to
explain why they as combinatorists ought to be interested in Con(ZFC).
You will find that it can't be done, in fact there is a huge gap in
understanding.

Joe, I think the real reason you profess not to appreciate Harvey's
result is very simple: you don't admit that Harvey's independent
finite combinatorial statements are combinatorial in a way that
Con(ZFC) is not.  This distinction is crucial.

> (Perhaps my objection is due to my reluctance to use words like
> "epochal" unless they are clearly called for.)

If that's the only objection, I will cheerfully retract my use of the
word "epochal" and replace it with something else.  Would you prefer
"earth-shattering"?  Or maybe, "cosmic"?  (Just kidding.)

-- Steve

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