FOM: Jan Mycielski's vs Harvey Friedman's use of large cardinals

[Martin Davis]

Dear fom-ers,

I'm off to Europe this week and likely away from email for six weeks. Here 
is a brief comment.

We had a discussion some time ago initiated by Joe Shoenfield on the issue 
of whether and in what sense Harvey's "necessary use of large cardinal" 
results are to be preferred to results of a straightforward 
metamathematical nature obtainable in a uniform manner and with much less 
trouble.

What's at issue isn't a matter to be settled by seeking a formal definition 
of "combinatorial" (as Joe had suggested) or by discussing how easy or hard 
it is to explain formal consistency to mathematicians with no logical 
training (as Jan does).

Harvey's theorems are intended to be recognizable by combinatorists as 
being the kinds of results they seek.  It has the additional intention of 
pushing the limits of what will be accepted by mathematicians as legitimate 
in proofs. (For a historical analogue, think of the struggles over general 
acceptance of AC.) For this to happen, it will be necessary for 
mathematicians to see that large cardinals are needed to obtain results 
they want to get.
Neither Jan's nor Joe's suggestions meet that test. Of course it is 
reasonable to discuss  to what extent Harvey's theorems approach that goal.

It is also worth noting that although metamathematical considerations do 
enter Harvey's work  in equivalences to 1-consistency, the PROOFS of those 
equivalences are intricate and difficult. I'd suggest that an appropriate 
challenge to someone who claims that these results are in spirit no 
different from the ones cited by Jan or Joe, is to show how to get Harvey's 
results from them in a straightforward way.







                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)
                        http://www.eipye.com