FOM: Harvey's `perfect' statement; Feferfest volume

Stephen G Simpson simpson at
Mon Feb 22 20:48:56 EST 1999

The background here is Harvey Friedman's program of finding
mathematically natural finite combinatorial statements which are
independent of strong systems.  Naysayers such as Shoenfield have
raised questions about how to interpret the phrase `mathematically
natural'.  Harvey's posting of 5 Feb 1999 02:46:52 gives the clearest
statement yet of what this phrase should mean and what kind of
combinatorial statement would be `perfect' in this sense.  Harvey
presents the following mathematicatically natural statement

  For all k there exists n so large that the following holds.  Let T
  be a finite rooted tree of height n in which each vertex has at most
  k immediate successors.  If T_i denotes T up to and including level
  i, then for some i there exists an inf-preserving embedding of T_i
  into T which maps level i into strictly higher levels.
and explains the senses in which this statement is `perfect'.

Unfortunately, the above statement is not independent of ZFC, but
`only' of Feferman's various systems of predicative mathematics
(actually it goes a bit beyond that).  Still, this independence result
is so nice in so many ways that I think it should be much more widely

Harvey, are you going to write up this `perfect' independence result
fairly soon?  If you do, then it occurs to me that you might
contribute the writeup to the Feferfest volume, because it seems to
fit in so well with so many of Sol Feferman's interests.  After all,
Feferman has returned to predicativity again and again over 35 years.
Also, in his recent survey paper `Does mathematics need new axioms?'
in the American Mathematical Monthly, Feferman mentions and critiques
some of the previous results in this specific line, viz. natural
finite combinatorial statements that are predicatively unprovable.  It
seems to me that your `pefect' independence result bears directly on
some of Feferman's more controversial comments in that article.

As you know from <>, I am
planning to at least mention your `perfect' independence result in my
own contribution to the Feferfest volume.  But this is a bit awkward
for me, and I think it would be much better if you could tell your own
story in that volume.

Feferfest editors, are you listening?

-- Steve

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