I now have a much clearer idea of just why this discussion can have this much traffic, yet actually have such limited productivity. Of course, I can simply continue to put up real time foundations, which seems to be working pretty well, and generate little interaction here. I was expecting a lot of interaction, and I am only beginning to understand why this is not likely to be the case.
Actually, the diagnosis of this is itself interesting. Fundamentally a foundationalist like me operates under a very different methodology than philosophers or mathematicians, or for that matter, mathematical logicians, including those with foundational or philosophical pretensions.
There is also the great silent majority on this email list who write nothing. So from time to time, I go offline and try to get some feedback on how I am doing. I get some responses, and all of them quite encouraging for me to continue. This leaves the majority of people from which I have no feedback yet. So right now, in the absence of negative feedback, I will continue.
I also have had plans to push the traffic back to Sol’s views as expressed in his manuscript. That CH is neither a mathematical nor a logical problem. This goes to the heart of the mathematical and logical status of higher set theory, something which has not been seriously discussed here. It is generally speaking an awkward topic since many people on this list, and in particular Hugh, Pen, Peter, Sy have made a living mostly under the premiss that higher set theory is a legitimate area of research mathematically and philosophically. In varying senses, neither Sol nor I accept this prima facie. Speaking for myself, I regard the status of higher set theory as a genuinely open issue, with supporting and non supporting arguments. I definitely work on both sides of the fence concerning varying forms of legitimacy of higher set theory. Or put differently, I definitely work on many sides of the fence on this.
A lot of the traffic concerns, in one way or another, “maximality in set theory”, and in particular a “program” being pushed, going under the flowery name of HP = hyperuniverse program.
This is being put forth as some sort of foundational program based on some idea of “intrinsic maximality of the set theoretic universe”.
The rationale of this program was questioned repeatedly – to varying degrees – by Hugh, Pen, Peter, and me.
One bottom line is that I recommended on the traffic that the name of the “program” be changed to CTMP = countable transitive models program. Then Hugh later endorsed my recommendation on the traffic.
The obvious reasons for this name change is that in fact the “program”, such as it is, is presented as a study of countable transitive models of ZFC. So it would simply be called CTMP if one is compelled to call it a program at all. But there is a foundational pretension being put forward, that it somehow supports a notion of “intrinsic maximality of the set theoretic universe”. As i have said repeatedly on the traffic, this simply cannot be the case. Countable models may be used as a major tool in establishing facts about some perhaps coherent formulation of “intrinsic maximality of the set theoretic universe”. But facts about countable models are not going to be the direct source of coherent treatments of anything like “intrinsic maximality of the set theoretic universe”.
With regard to “intrinsic maximality of the set theoretic universe”, the traffic is interesting. There are some unexpected features.
- My interest was piqued by the use of “intrinsic maximality”. I have long thought that intrinsic justifications, generally, are extremely important – even essential. And there are all sorts of great opportunities and challenges to uncover new intrinsic justifications in a wide variety of foundational arenas – not just maximality.
- I was surprised to learn that Pen and Peter both are so highly skeptical of “intrinsic justifications” specifically with regard to set theoretic axioms, and perhaps more generally as well. I am not sure of Hugh’s position on this. Why was I surprised? I knew that they like to emphasize “extrinsic justifications”, but I had thought that they still endorsed “intrinsic justifications” to a fair extent. Another reason I was surprised, was implicit. I believe that the usual discussion of “extrinsic reasons for set theory” is deeply flawed and represents a lack of acknowledgement of many key features in mathematics generally, and many key attitudes of mathematicians. Specifically, there is a kind of RESTRICT! or DON’T MAXIMIZE! that is going on pervasively – at some important level – all through mathematics and with mathematicians generally. The “extrinsic/maximize” proponents (including Pen and Peter) surely have a defense against this loud attack. They can try to draw a distinction between what mathematics and mathematicians want to RESTRICT! and what would amount to restriction in set theory such as restricting to L. And then the debate goes on, with deeper issues as to the very point of what mathematics is and what higher set theory is, taking front and center. I am up for this debate, but I have as of yet no indication that Pen and Peter and others are up for this debate.
- Looking at the major gap between something like the original IMH (inner model hypothesis) and any kind of truly coherent presentation of “intrinsic maximality”, I continually tried to get a truly coherent presentation of just what “intrinsic maximality” we are talking about, in truly fundamental terms. I was completely ignored perhaps a dozen times in one way or another. Now I see why I was ignored by Peter and Pen on this – because they do not advocate any relevant kind of “intrinsic maximality of the set theoretic universe” at all. But I directly addressed Sy on this, as he is strongly advocating “intrinsic maximality of the set theoretic universe”, and he is not even acknowledging my requests for a deliberate and careful discussion of just what “intrinsic maximality of the set theoretic universe” is supposed to mean, or why it does not deserve interactive discussion on this thread, is highly unprofessional.
- Since people do not normally openly act in highly unprofessional ways in front of dozens of people, there has to be an explanation. The explanation is, of course, that Sy is no danger of being viewed as highly unprofessional, because “everybody will know that there must be something personal going on”. Actually, there is nothing personal going on, as far as I am concerned. I am simply treating Sy and always have as a legitimate member of the profession. His ideas are as good as many people’s, and his ideas are as bad as many people’s. I don’t discriminate.
- Since all of the traffic, except Sol and me, took ZFC “for granted”, I automatically assumed that the prevailing view here was that ZFC was intrinsically justified – and specifically, intrinsically justified by “maximality of the set theoretic universe”. I have long been interested and have foundational programs concerning trying to “justify ZFC intrinsically” – including as a transfer from the finite (Transfer Program). So now I see that probably Pen and Peter do not subscribe to an “intrinsic story for ZFC”. Even more startling was Sy’s being dubious about AxC being “intrinsically justified by set theoretic maximality”. This from somebody pushing a misnamed “program” HP put forth as being responsive to “intrinsic set theoretic maximality”!
- So I got interested in trying to give an “intrinsic maximality” justification for AxC. I have just got started with this, but already there has been a very attractive spinoff subject that I recently wrote about. I am hoping for a lot more spinoff subjects to come, even if I don’t succeed in getting an intrinsic maximality justification for AxC. In fact, more broadly, the topic of giving intrinsic justifications for ZFC is extremely attractive. This includes the possibility of proving that there are no intrinsic justifications for ZFC of certain kinds. At some point soon, I will leverage off of ZFC, and look at intrinsic justifications for “there exists a nonconstructible real”, especially through “intrinsic maximality”. Again also looking for negative results indicating that this cannot be intrinsically justified. Then I plan to start talking hopefully systematically and fundamentally about CH.
- In my continuation, I will start by reviewing the spinoff subject already generated concerning AxC that I have previously discussed.