Re: Paper and slides on indefiniteness of CH

On Sun, Oct 19, 2014 at 5:06 AM, Radek Honzik wrote: Dear all,

I will attempt to answer briefly the questions posted by Harvey. My view on HP is different from Sy’s, but I see HP as a legitimate foundational program.

The most accomplished contributors to this list seem to be doubtful.

Thanks for your replying. Your reply doesn’t contain a mish mash of hidden assumptions, not so hidden assumptions, question begging, ignoring criticisms, missed opportunities for joining issues, evasions, crude bragging about important busy activities, undefined terms, total lack of respect for the audience who does not keep up with specialist jargon, and a long list of other sins that make this thread an example of how not to do or even talk about foundations and philosophy.

I appreciate that you have for the most part avoided these sins.

0] At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place?

Let me write IMST instead of “(intrinsic) maximality in set theory” for the sake of brevity.

I doubt IMST can mean more than “viewing sets as big as possible, without the use of considerations based on practice of set theory as the main incentive”.

I think you mean “viewing the set theoretic universe as inclusive as possible”?

Your sentence has an indirect construction that really is not necessary and slows down the reader.

“Intrinsic” is thus temporarily reduced to “non-extrinsic”; in view of the heavy philosophical discussions around this notion, I prefer to give it this more restrictive meaning. Note that “extrinsic”, unlike “intrinsic”, has a well-defined inter-subjective meaning. This leaves us with the word “big”; I guess that this is the primitive term, which cannot be defined by anything more simple — at least on the level of general discussion.

Again, I don’t think you want to use “big” here – I am suggesting something very similar – the set theoretic universe is as inclusive as possible. I regard this as only an informal starting point and the job is to systematically explore analyses of it.

Admittedly, this definition is far from informative. For me, HP is a way of explicating this definition in a mathematical framework. Making its meaning more precise, and by the same token, less general. A discussion should be if other approaches — which set out to get real mathematical results — retain more of the general meaning of the term IMST. No approach can retain all the meaning of ISMT because it is by definition vague and subjective; thus HP should not be expected to do that.

But there is the real possibility of saying something generally understandable, surprising, and robust. I haven’t seen anything like that in CTMP (aka HP).

1] Why doesn’t HP carry the obvious name CTMP = countable transitive model program.

Because the program was formulated by Sy with the aim of having wider application than the study of ctm’s.

Since the name “hyperuniverse” specifically refers to the countable transitive models of ZFC, period, it amount to nothing more than a propogandistic slogan designed to lure the listener into thinking that there is something profound going on having to do with the foundations of set theory. But since nothing yet has come out of this special study of ctms for foundations of set theory, even propogandistic slogans about ctms are premature.

2] What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”?

Countable models are a way of explicating IMST. It is a technical convenience which allows us to use model-theoretic techniques, not available for higher cardinalities.

What you have here is an unanalyzed idea of “intrinsic maximality in set theory”, and before that is analyzed to any depth, you have the blanket assumption that countable transitive models are going to be the way you can formulate what is going to become the analysis of “intrrinsic maximality in set theory”. The real agenda is a creative or novel analysis of “intrinsic maximality in set theory”, and BEFORE that is accomplished any “proof” that ctms will do by some sort of Lowenheim Skolem argument is bogus. Of course, you can set up some idiosyncratic framework, pretend that you going to make this your analysis of “intrinsic maximality in set theory”, and then cite Lowenheim Skolem. But that is bogus. After you make some creative or novel analysis, and work through the problematic issues (inconsistencies and other non robustness arising out of parameters, sets of sentences, etcetera), and have a framework that is credible and well argued, you can cite the Lowenheim Skolem theorem – if it really does apply correctly – to say that any claims of a certain form are equivalent to claims of the form with ctms. That would make some sense, but I still would not advise it since the proper framework, if there is any, is not going to be based on ctms. Ctms would only be a convenience.

This is the serious conceptual error being made in endless emails by Sy trying to justify the use of ctms. An unjustified framework for treating “intrinsic maximality in set theory” is alluded to, and to the extent that it is precise, one quotes Skolem Lowenheim to argue that one can wlog work with ctms. This is a very serious question begging sin. The issue at hand is first to have a novel or creative and well argued and thought out framework for treating “intrinsic maximality in set theory”. AFTER THAT, one can talk about the convenience – but NOT the fundamental nature of – using ctms.

This reversal of proper order of ideas – putting the cart before the horse – is a major error in work in foundations and philosophy.

IN ADDITION, on the mathematical level, I quote from Hugh. This indicates that even in frameworks proposed beyond IMH, there is no Lowenheim Skolem argument, and one is compelled to make the move that ctms are fundamental, rather than just a convenience. Here is the exchange:

Sy wrote:

More details: Take the IMH as an example. It is expressible in V-logic. And V-logic is first-order over the least admissible (Goedel-) lengthening of V (i.e. we go far enough in the L-hierarchy built over V until we get a model > of KP). We apply LS to this admissible lengthening, that’s all.

Hugh wrote

This is of course fine for IMH. But this does not work at all for $\textsf{SIMH}^\#$. One really seems to need the hyperuniverse for that. Details: $\textsf{SIMH}^\#$ is not in general a first order property of $M$ in $L(M)$ or even in $L(N,U)$ where $(N,U)$ witnesses that $M$ is #-generated.

MY COMMENT: So we may be already seeing that in some of these approaches being offered, one must buy into the fundamental appropriateness of ctms in the philosophy, and not just an automatic freebie from the Lowenheim Skolem theorem. FURTHERMORE, IMH, where reduction to ctm makes sense through Skolem Lowenheim, has not even been seriously analyzed as an “intrinsic maximality in set theory” by serious foundational and philosophical standards. There is a large array of issues, including inconsistencies and non robustness involving parameters and sets of sentences, and so forth.

Aside: I do not quite understand why the discussion rests so heavily on this issue: everyone seems to accept it readily when we talk about forcing (I know it can be eleminated in forcing, but the intuition — see Cohen’s book — comes from countable models). Would it make a difference if the models had cardinality omega_1, or omega and omega_1, or should they be proper classes etc? Larger cardinalities would introduce technical problems which are inessential for the aims of HP.

The crucial issue can be raised as follows. Do we or do we not want to take the structure of ctms as somehow reflecting on the structure of the actual set theoretic universe?

I am interested in seeing what happens under both answers. What is totally unacceptable is to make the hidden assumption of “yes we do” while pretending “no we are not because of the Löwenheim Skolem theorem”. That is just bad foundations and philosophy.

I am going to explore what happens when we UNAPOLOGETICALLY say “we do”. No bogus Löwenheim Skolem.

3] Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why?

IMST by historical consensus includes at this moment ZFC. “Historical consensus” for me means that many people decided that the vague meaning of IMST extends to ZFC. I do think that this depends on time (take the example of AC). HP is a way to raise some new first-order sentences as candidates for this extension.

Then what is all this talk on the traffic doubting whether AxC is supported by “intrinsic maximality of the set theoretic universe?”

4] What is your preferred precise formulation of IMH? E.g., is it in terms of countable models?

Yes.

5] What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”?

I view the process of obtaining results in HP like an experiment in explicating the vague meaning of IMST. It is to be expected that some of the results will be surprising, and will require interpretation.

This is a good attitude. However, there is still not much that has come out, and it is still unclear whether this will change. So declaring it a “program” without having the right kind of ideas in hand, and coining a jargon name, is way premature.

6] What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities.

It is a creative process: explicate IMST by principle $P_1$ — after some mathematical work, it outputs varphi (such as $P_$1 = IMH, $\varphi$ = no inaccessible). Then try $P_2$, etc ($P_2$ can be a “redesigned”, or “modified” version of $P_1$). Of course, one hopes that his/her understanding of set theory will be helpful in identifying P’s which have potential to output nice (good, deep) mathematics. It is essential that the principles $P$‘s should be as practice-independent as possible (= intrinsic, in my reading); that is what makes the program foundational (again, in my more narrow sense).

Taking into account what you are saying, and the difficulties that Hugh has been pointing out about the post IMH proposals, this does not have enough of the features of a legitimate foundational program at this stage. It has the features of a legitimate exploratory project without a flowery name and pretentious philosophy. We don’t know if it is going to develop into a legitimate foundational program, which would justify flowery names and pretentious philosophy.

Harvey