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

Solomon Feferman sf at Csli.Stanford.EDU
Sun Mar 15 22:28:48 EST 1998

  In his posting of 11 March 11:36, Friedman stated results about finite
trees for which he tells us that
1. They are provable in ZFC + the existence, for all k, of k-subtle
2. They are not provable in ZFC + the existence of k-subtle cardinals, for
each specific k for which this theory is consistent.
  In his posting of 12 March 13:25, Simpson declared this to be an epochal
advance.  In response, Franzen (13 March 14:19) rightly questioned this
premature declaration (some would call it "drum-beating" or "advance
hype").  In particular, he said:
  "So without in any way seeking to belittle what is surely a remarkable
piece of work, I think it's a bit too soon to characterize it as
tremendously important progress in f.o.m."
  I agree fully.  Others have disagreed, including Davis and Tait, in
addition to Simpson and Friedman.  The latter two have pounced on Franzen
for not trying to understand the results.  Franzen questioned how this fit
with the "general interest line" promoted previously by Friedman as what
are the most important things to work on.  Friedman says its enough if
competent specialists (combinatorists) have found it to be of interest and
that large cardinal axioms are necessary.
  In all this, there has not been any mention of what I consider to be the
principal foundational question.  If these results 1 and 2 are anything
like previous such results by Friedman and others, 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.  I have to
reiterate from earlier postings of mine:
3. These do not show that such large cardinal principles are necessary as
first-class mathematical principles.
4. 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 can think of plausibility arguments for such beliefs, but they do not
count as even that minimal kind of justification.  Without additional
argument, as things stand, the purported necessary use of such large
cardinal principles is simply begging the question.

Sol Feferman

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