FOM: Friedman's independence results, an epochal f.o.m. advance

Harvey Friedman friedman at
Fri Mar 13 13:14:33 EST 1998

Reply to Franzen 2:19PM 3/13/98:

>  Steve Simpson says:
>  >I would like to call attention to Harvey Friedman's posting
>  >       FOM: 12:Finite trees/large cardinals
>  >of 11 Mar 1998 11:36:36.  It represents tremendously important
>  >progress in f.o.m.
>  This may well be the case, but remains to be established, at least
>if f.o.m. is to be a subject of interest to non-specialists.
>  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.

The combinatorial Propositions represent straightforward combinations of
classical Ramsey theory and standard setups for inserting new vertices into
finite trees. They are a successor to a group of 4 earlier Propositions
which have been the subject of a second year undergraduate course at UCSD
for computer science majors. That is why I included the url's and which presumably you
have consulted. These earlier Propositions were presented by people at UCSD
to approximately 25 researchers in combinatorics and computer science, and
were generally regarded as interesting, natural, basic, and simple.

The later Propositions discussed in my posting of 11:36AM 3/1198 were
presented by me, personally, to a variety of combinatorists, and
combinatorial computer scientists, including some of the world's leading
experts in Ramsey theory. Of the four formulations in my posting, several
of these people had their own favorites, and all of them regarded their
favorite one as interesting, natural, basic, and simple.

>What is needed to convince people
>that these are "very natural combinatorial propositions" is to find
>some striking applications of them. Even I could no doubt grasp
>these principles if I set my mind to it, but I need some incentive.

I already have convinced people; one of the leading experts in Ramsey
theory needed no convincing. The results represent an area of
discrete/finite combinatorics which is sufficiently interesting in its own
right. The four Propositions in the above url's came about as applications
of yet another earlier Proposition requiring large cardinals, in a joint
effort with me and a researcher at UCSD. Also I have been told of some
further applications, although I have not yet seen the details.

Having said this, I am still - and always have been for 10-30 years - in
the process of trying to do better. I have some yet simpler forms of these
results which I conjecture require large cardinals in the same sense.

I will be lecturing on these results later in the year, and I expect that
in a lecture context, these Propositions will be trivially understood by
the relevant audiences. It may look somewhat complicated on e-mail, without
decent looking symbols, especially if the reader is suspicious.

>  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.

It is not a "significant" circumstance. The subtle cardinal hierarchy is
completely natural and well known among set theorists, and well understood
since 1973 (Baumgartner). The hierarchy lies above the more often quoted
weakly compact cardinals and indescribable cardinals, and below the more
often quoted kappa arrows omega. I have a manuscript with new, very simple
definitions of this hierarchy. Of course, if one only understands, e.g.,
inaccessible cardinals and measurable cardinals, one can weaken the result
to say that the Propositions can be proved in ZFC + "there exists a
measurable cardinal," but not in ZFC + "there exists an inaccessiable

>  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.

"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.

I suggest that it is more reasonable for you to be asking questions rather
than telling us about how complicated you find something, or how perplexed
you are, or how perplexed you are that anybody would not be perplexed, or
making judgments as to what is needed to convince people, or when what
should be characterized in what way.

For instance, you could ask the following:

1. Have you (Friedman) gotten feedback from combinatorists or computer
scientists as to how natural they find these Propositions?
2. What are subtle cardinals, and k-subtle cardinals, and how do they
compare with other large cardinals?

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