FOM: cardinals and finite statements

[Harvey Friedman]

Reply to Feferman 7:28PM 3/15/98:

>  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
>cardinals.
>2. They are not provable in ZFC + the existence of k-subtle cardinals, for
>each specific k for which this theory is consistent.

In fact, I proved, within (weak fragments of) ZFC that they imply the
consistency of ZFC + {there exists a k-subtle cardinal}_k. As a Corollary,
one knows that it is necessary to use large cardinals in order to prove
them in a precise sense. More specifically,

COROLLARY. Any extension of ZFC that proves these statements is an
extension of ZFC in which ZFC + {there exists a k-subtle cardinal}_k is
interpretable.

This is obviously weaker than the demonstrably false assertion:

COROLLARY (false). Any extension of ZFC that proves these statements is an
extension of ZFC in which ZFC + {there is a k-subtle cardinal}_k is
provable.

But it nevertheless reasonably counts as a necessary use of large cardinals
in light of the Godel incompleteness phenomena.

>  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 have never openly declared this to be an epochal advance, or any kind of
advance at all. I concede that I am not particularly anxious to disuade
anyone from saying this sort of thing, though. But I think that the fom
should realize, and many already do, that Feferman is affectionately called
"the wet blanket of foundations." I once asked Feferman whether Cohen's
work in set theory was significant. He responded "It is certainly striking.
But we still don't know what it's significance is." And this is after 35
years. They have been colleagues for even longer!

>  I agree fully.  Others have disagreed, including Davis and Tait, in
>addition to Simpson and Friedman.

Wait a minute. I didn't say anything of this kind.

>The latter two have pounced on Franzen
>for not trying to understand the results.

For someone who is usually very very careful with words, this is very
disappointing. Since you use Franzen's impulsive posting of 2:19PM 3/13/98
as ammunition for your wet blanket, I feel that I must properly state what
I said about his posting.

I was complaining that Franzen openly declared on the fom that he was
shocked that "anybody can make sense of them," thereby giving people who
have not tried to make sense of them a very negative impression; in fact,
biasing them. Since one essential point about these Propositions is their
clarity and naturalness - which is considerable - this is tantamount to a
public judgement. And I was complaining that instead of finding out what
combinatorists think of this combinatorial statement, Franzen decides to
tell the fom what is needed to "convince people that these are "very
natural combinatorial propositions"". And then I complained that instead of
trying to find out what a k-subtle cardinal is in a scholary way, he
conducts a "search" on AltaVista, coming up empty, and then advises the fom
as to what is needed to do about the existence of subtle cardinals. The
search on AltaVista was apparently inept, since I know that subtle
cardinals can be found with standard search engines in less than 15
seconds. Here is most of what Franzen wrote 2:19PM 3/13/98:

>  My own tendency as I attempt to penetrate the combinatorial
>principles here at issue is to lapse into slack-jawed wonder that
>anybody can make sense of them, let alone formulate them. ... What is
>needed to convince people
>that these are "very natural combinatorial propositions" is to find
>some striking applications of them. ...
>
>  Also, it is a significant circumstance that the only occurrence of
>the phrase "subtle cardinal" on any web page indexed by AltaVista is a
>reference to Cardinal Granvelle. To establish Friedman's results as
>important progress in f.o.m., a principle that yields the existence of
>subtle cardinals must be established as a comprehensible and
>potentially acceptable addition to the axioms of set theory.

And my responses to him are in 7:14PM 3/13/98 and 10:25AM 3/15/98.

>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.

Specifically, I wrote in 10:25AM 3/15/98:

"A coherent body of discrete and finite combinatorial results, regarded as
interesting, natural, basic, and simple by relevant practitioners, has been
discovered and shown to be provable only by going well beyond the usual
axioms of mathematics via standard axioms of higher infinities." A
non-specialist can understand this finding.

This handles the general intellectual interest issue rather nicely. I
certainly would not have spent approximately 30 years on this issue unless
I was completely convinced of its general intellectual interest in such
terms.

>  In all this, there has not been any mention of what I consider to be the
>principal foundational question.

No. It's another important 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.

This is merely a totally idiosyncratic remark of Feferman's that no one
else would make, which I now dispense with. First of all, instead of trying
to dampen the reception for the results by talking about previous work of
mine and others, Feferman should have at least referenced the work of mine
and others. Feferman is saying:

"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."

Now, would Feferman say the following?

"They merely show that the combinatorial results in question are equivalent
to the 1-consistency of ZFC." Or

"They merely show that the combinatorial results in question are equivalent
to the 1-consistency of Z with bounded separation."

Yet as Feferman knows, I have been claiming, or at least conjecturing, that

"the combinatorial results in question can be naturally weakened in order
to become equivalent to the 1-consistency of the formal system
corresponding to any natural level of the cumulative hierarchy, from
V(omega) up to V(kappa), where kappa is the limit over k of the first
k-subtle cardinal."

Thus not only do we know that there is a correspondence between some
standard axioms of higher infinity and appropriately significant
discrete/finite combinatorial problems, we actually know that there is a
one-one correspondence between natural levels of the cumulative hierarchy
(up to the level we are talking about) and appropriately significant
discrete/finite combinatorial problems. Various people may have different
attitudes about the various levels of the cumulative hierarchy - in terms
of their existence or coherence or consistency or whatever. Take the level
anyone is concerned with at any moment for any purpose (bounded by the
subtle cardinal hierarchy). There is a corresponding appropriately
significant discrete/finite combinatorial problem.

Granted, the technical work has not been completed for publication, and the
work involved is sufficiently complicated that it needs to be extremely
carefully written up before being really claimed. But it will be. At least
these papers are done: Finite functions and the necessary use of large
cardinals, to appear, Annals of Math. And the paper under discussion,
Finite trees and the necessary use of large cardinals.

Here are some conjectures that I have confidence in, but must fall short of
claiming:

PROPOSITION 1. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set are
vertices with the same number of ancestors.

PROPOSITION 2. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set with
the same first k-1 elements are vertices with the same entirely lower
ancestors.

PROPOSITION 3. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set with
the same min are vertices with the same entirely lower ancestors.

PROPOSITION 4. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set are
vertices with the same entirely lower ancestors.

1 and 4 are from the paper. In the paper, 4 was proved from ZFC + (for all
k)(there exists a k-subtle cardinal). And in the paper, 4 was proved to
imply the consistency of ZFC + {there exists a k-subtle cardinal}_k. Also 1
was proved in the paper from Z. I know how to prove 2 in ZFC + there exists
an inaccessible cardinal. I know how to prove 3 in ZFC + "for all k, there
exists a k-Mahlo cardinal."

CONJECTURES. 1 is provably equivalent to the 1-consistency of Z with
bounded separation (or type theory). 2 is provably trapped between the
1-consistency of ZFC and the 1-consistency of MKC. 3 is provably equivalent
to the 1-consistency of ZFC + {there exists a k-Mahlo cardinal}_k. 4 is
provably equivaletn to the 1-consistency of ZFC + {there exists a k-subtle
cardinal}_k.

This is just a sample. Basically the situtation is this. We already know a
lot about how to adjust transfinite Ramsey theory in order to hit a lot of
natural cardinals right on the head. What I can do is pull them down into
Rmasey theory on the INTEGERS, and hit those same natural cardinals right
on the head (up to 1-consistency).

>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.

This is a separate issue that remains to be played out. Are you completely
disavowing Godel's point of view, where he said that principles such as
these are to be accepted as they gain consequences?

>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.

This is a separate issue that remains to be played out. Again: are you
completely disavowing Godel's point of view, where he said that principles
such as these are to be accepted as they gain consequences?

>  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.

You have merely raised a separate issue in f.o.m.: namely, why should we
believe in large cardinals or their consistency? I have claimed only the
following:

A. The issue you raise is joined in a much more serious way than ever
before, with the establishment of a one-one correspondence between
appropriately natural discrete/finite combinatorial statements and the
various natural levels of the cumulative hierarchy (up to quite a high
level). Some of this needs to be carefully documented, of course.

B. The results fit in exactly according to what Godel had in mind from his
writings. He regarded such results as reasons for the acceptance of new
axioms.

C. For work more directly connected with the separate issue in f.o.m. that
you raise, there is my work on transfer principles and also on new
axiomatizations of set theory with 2 universes.

I would like to see the wet blanket dry out.