Re: Paper and slides on indefiniteness of CH

Dear All,

Thanks, first, to Peter Koellner, for his contribution to the discussion.  I will only address the first part of it concerning the relation to the EFI Project of my paper that initiated the present discussion here.  My draft paper for the EFI project to which Peter refers was entitled “Is the Continuum Hypothesis a definite problem?”, let’s refer to it as (F1); the draft paper that initiated this discussion is entitled “The continuum hypothesis is neither a definite mathematical problem nor a definite logical problem”; call it (F2). [Links to both of these can be found on my home page at #95 and #115, resp.] In my view, (F2) is a considerable revision of (F1), so last spring I sent it to a group of colleagues for comments.

That was not intended to open an online discussion and it only became such when Sy Friedman made an email response to (F2) with cc’s to all those on my list that he expanded to include some further names; since then others have added still more names.  It is not at all what I expected or was initially seeking, namely individual comments on (F2) to me alone.  In the process, the discussion widened considerably and following it has indeed become something like “engaging with a hydra.”  Speaking for myself, I have already benefited from it w.r.t. my original purpose even though I have not been able to find time to absorb all the ins and outs of the discussion. In the meantime, I still welcome any individual personal comments on (F2) that people might like to make without getting involved in the online discussion.

The format of the EFI Project to which Peter refers consisted of a series of individual lectures at Harvard over the academic year 2011-2012.  The discussion of each was primarily limited to the faculty and students attending the given lecture, though online exchange was solicited.  There was a short meeting over the Labor Day weekend of 2013 to which all the speakers were invited; unfortunately, neither Tony Martin nor I could attend, so I cannot report on the discussion there.  Peter reports a great deal of convergence of views on the “search for new axioms”.  I wonder about that specifically with respect to the status of CH, since, e.g. under the multiverse approaches of Hamkins and Steel among others, it has no absolute status.  In any case, it seems to me that the discussion here shows that there is a greater divergence of views in our community than reflected in the results of the EFI project.

This is not to diminish at all the importance of the EFI project.  I think this was very important and we have Peter to thank for organizing it and for his critical examination of various of the contributions, mine among them. That required a great deal of energy, time and thought for which we should be very grateful.  And we should very much welcome the prospect of seeing “a final volume containing the papers, commentary, and accounts of the exchanges.”

Peter has provided a link to his acute criticisms of (F1) and says that they apply equally well to (F2). However, (F2) was revised to meet  those criticisms (evidently not to his satisfaction) and contains some essentially new and, in my view, quite important points not addressed in them.  A certain amount of that has emerged in our ongoing online discussion and I will not go over them again here. Instead, I hope those of you who are interested will judge all this for yourself.

In the second part of Peter’s message, not duplicated below, he has also raised specific questions about the material of the Appendix to (F2), material which did not occur at all in (F1).  I shall respond to those separately when I have more time.


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