Obviously Pen and Sy are sort of talking past each other, which is understandable given the abstract nature of a discussion on general strategy of research in set theory with almost no mathematical specifics presented. In order to help the discussion, I have a few suggestions.
- Woodin and other close associates like John Steel have definite specific proposals for “settling” CH through explicit or reasonably definite conjectures. The statements are very complex, even for most set theorists, let alone logicians generally, or philosophers.
- Sy has raised various kinds of objections to their proposals for “settling” CH. Sy has been offering an alternative plan for “settling” CH. Sy is claiming some substantial advantages of his plan over Woodin/Steel plans. In particular, either explicitly or by inference, Sy is claiming that his plans are comparatively straightforward.
- I was intrigued by the offering up of a comparatively straightforward plan to “settle” CH. So I asked for Sy to provide some account of these comparatively straightforward plans here in this forum, so people can comment on them. I was hoping that this would not only help the present discussion, but also bring in some other people to comment on the advantages and disadvantages of Woodin/Steel versus Sy et al. I don’t know why I have been ignored. Sy?
Now here are some other matters related to the discussion.
- People seem to be blackboxing the idea of “good mathematics” or “good set theory.” These notions are desperately in need of some sort of elucidation particularly by people with foundational sensibilities. My working definition during my entire career has been “mathematics with a clear foundational purpose”. I do recognize that there can be “good mathematics” that does not – at least not obviously – fit that criterion. This is a crucial issue – what is good math or good set theory – as on at least one kind of reading, almost none of it has any clear foundational purpose.
- Sol’s use of the phrase “mental picture” in his earlier email reminded me of some ideas that I had left up in the air for some time. I buckled down and wrote the following posting to the FOM email list. I close with a copy of the substantive part.
I have an extended abstract on some mental pictures, which can be used to arguably justify the consistency of certain formal systems.
SOME MENTAL PICTURES I
by Harvey M. Friedman*
August 11, 2014
*This research was partially supported by the John Templeton Foundation grant ID #36297. The opinions expressed here are those of the author and do not necessarily reflect the views of the John Templeton Foundation.
Abstract. All mathematicians rely on mental pictures of structures. Some can be used to offer justifications for certain axiom systems. Here we use them to make arguable justifications ranging from Zermelo set theory to ZFC to various large cardinal hypotheses.