FOM: What is necessary about the use of large cardinals?

[Stephen G Simpson]

Discussing Harvey Friedman's results about the necessary use of large
cardinals to prove certain combinatorial theorems about finite trees,
Solomon Feferman 15 Mar 1998 19:28:48 writes:

 > they merely show that the combinatorial results in question are
 > equivalent to the 1-consistency of ZFC + the existence, for all k,
 > of k-subtle cardinals.

"Merely"?  What more could you reasonably expect?

 > These do not show that such large cardinal principles
 > are necessary as first-class mathematical principles.

I'm not sure what you mean by "first-class".  

What Harvey proves is that (1) the 1-consistency of the large
cardinals is *logically necessary*, in fact necessary and sufficient,
to obtain the combinatorial theorems in question.  Moreover: (2) we
don't know of any any coherent f.o.m. picture which implies the
1-consistency of the large cardinals, other than the large cardinal
picture itself.

I think it's reasonable to interpret (1) and (2) together as implying
that the the large cardinal picture is *logically necessary* to obtain
the combinatorial theorems, in the present state of knowledge.  And
nobody has any idea how to transcend this present state.

 > What needs to be argued is why it is even necessary, given 1 and 2,
 > to believe in the 1-consistency of such principles over ZFC.

I don't think that's the point.  This is a different and inappropriate
sense of "necessary".  Nobody wants to force anybody to believe in or
not believe in the 1-consistency of large cardinals.

I think that the new insight in this work of Harvey can be formulated
as follows: If mathematicians want to solve the finite combinatorial
problems which are answered by Harvey's combinatorial statements,
there is no way to do so without examining what hitherto might have
been regarded as remote philosophical or f.o.m. questions, concerning
large cardinals.  In other words, large cardinals are intertwined with
mathematical practice in a previously unknown way.

I do think this represents tremendous f.o.m. progress.

 > Without additional argument, as things stand, the purported
 > necessary use of such large cardinal principles is simply begging
 > the question.

"Begging the question"?  I find this much too harsh.  Question-begging
is a classical logical fallacy.  Are you accusing Harvey of committing
a classical logical fallacy?  I hope not.

On the whole, I think you are underestimating the magnitude of this
advance.

-- Steve