[FOM] 408: Kernel Tower Theory 2

[Timothy Y. Chow]

Jan Pax wrote:

>I believe that most FOMers do not know and would like to know how your 
>extremely interesting THEOREMS involving consistency of various large 
>cardinals over weak base theories can be proved. If you put effort into 
>defining notions and stating theorems, please give us, or promise to give 
>in the future, at least very rough feeling about the ways of PROOFS. Any 
>single proof would suffice.

I'm not sure exactly what you're asking for here, but if you're looking 
for some insight into the link between the consistency of large cardinals 
and various "purely combinatorial" Pi-0-1 statements, then you could look 
at his paper "Finite functions and the necessary use of large cardinals," 
Ann. Math. 148 (1998), 803-893, or at the downloadable preprint of his 
book on Boolean Relation Theory, available on his website.

At a very high level, you can already see a hint of the connection when 
you observe that the very definition of many large cardinals is just some 
highly combinatorial statement about well orders.  In slightly more 
detail, here's an illuminating paragraph from Harvey's aforementioned 
Annals paper:

   The general strategy for using large cardinals in the integers
   is as follows.  We start with a discrete (or finite) structure
   X obeying certain hypotheses H.  We wish to prove that a certain
   kind of finite configuration occurs in X, assuming that H holds.
   We define a suitable concept of completion in the context of arbitrary
   linearly ordered sets.  We verify that if X has a completion with the
   desired kind of finite configuration, then X already has the desired
   kind of finite configuration.  We then show, using hypotheses H,
   that X has completions of every well-ordered type.  We now use the
   existence of a suitably large cardinal lambda.  Using large cardinal
   combinatorics, we show that in any completion of order type lambgda,
   the desired kind of finite configuration exists.  Hence the desired
   kind of finite configuration already exists in X.

It's also worth mentioning that proving that Friedman's propositions 
follow from the existence of large cardinals is usually much easier than 
proving that said propositions are unprovable in ZFC + a slightly smaller 
large cardinal.  In this regard, they resemble Paris-Harrington.

Tim