# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 19, 2014, at 7:01 PM, Sy David Friedman wrote:

Dear Hugh (likely my last mail for a while, due to my California trip),

(Fine let’s call it HP and not CTMP). HP seems now to be a one-principle program ($\textsf{SIMH}^\#$).

This is nonsense. One may bring in cardinal maximality, inner model reflection or various forms of unreachability as well. You have an $\textsf{SIMH}^\#$ fixation.

I am fixated on $\textsf{SIMH}^\#$ because you have provided nothing else at this stage to examine which is precise. Give me a property true of the preferred universes according to HP now, other than $\textsf{SIMH}^\#$, to think about.

You just wrote to Pen:

Yes, but what is new is to use a multiverse as a tool to gain knowledge about V.

How about one example of a sentence which HP currently declares as true about V and which is “new”. If you cannot provide such a sentence then what is the sense of what you wrote to Pen? Perhaps what you meant to say is “Yes, but what is possibly new…”

My difficulty is this. You present HP both as a program which has discovered new and specific insights about V and as a program in its infancy for which such demands are unfair at this stage.

Ok, which is it?

# Re: Paper and slides on indefiniteness of CH

Dear Hugh (likely my last mail for a while, due to my California trip),

On Sun, 19 Oct 2014, W Hugh Woodin 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.

This is of course fine for IMH. But this does not work at all for SIMH#. One really seems to need the hyperuniverse for that.

Details: 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.

You are of course right and I have been suppressing the technical details required to deal with this to avoid complicating the discussion. The point is that with any property that refers to “thickenings” one must make use of V-logic, let’s call it M-logic to match your notation. Then what I have been suppressing is that #-generation is to be taken as the consistency in M-logic (extended with new axioms making the ordinals standard) of the obvious theory expressing the iterability of a presharp that generates M. LS can be applied because any presharp that embeds into an iterable presharp is iterable. Handling variants of the $\textsf{IMH}^\#$ takes more work, but can be done. Of course the difficulty is that we are not dealing with actual objects but with the consistency of theories. I’ll write more about this when I get a chance.

Of course if one is happy to adopt the Hyperuniverse from the start (without the “reduction”) then these technical issues disappear.

(Fine let’s call it HP and not CTMP). HP seems now to be a one-principle program ($\textsf{SIMH}^\#$).

This is nonsense. One may bring in cardinal maximality, inner model reflection or various forms of unreachability as well. You have an $\textsf{SIMH}^\#$ fixation.

Further progress seems to require at the very least, understanding the implications of $\textsf{SIMH}^\#$.

There could be progress with other criteria in the meantime.

As I said in my last email on this, it is impossible to have a mathematical discussion (now) of $\textsf{SIMH}^\#$ since it has been formulated so that one cannot do anything with it without first solving a number of problems which look extremely difficult. And I am not even talking about the consistency problem of $\textsf{SIMH}^\#$.

Just to be clear: This is not a criticism of $\textsf{SIMH}^\#$, it just saying it is premature to have a mathematically oriented discussion of it, and therefore of HP.

I partly agree: The important issues now concern the formulation of the HP, not the details of the mathematical analysis of the maximality criteria.

So (and this is why I have repeated myself above), I do not yet draw the distinction you do on HP versus the study of ctm’s because there not yet enough mathematical data for me to do this.

This is quite ridiculous. The study of ctm’s is obviously much, much broader than the specific types of questions relevant to the HP (maximality criteria).

Hopefully this situation will clarify as more data becomes available.

That is what I have been trying, without success until now, to say to you for several weeks already. The important task for now is to just recognise the legitimacy of the approach, not to evaluate the results of the programme, which will take time.

Have a productive week in California!

Thanks!

Concerning the future of HP, I will make a prediction: HP ends up with PD.

This has already almost happened a number of times (strong unreadability and the unreachability of V by HOD).

I think this very likely, plus a lot more!

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 19, 2014, at 5:25 AM, Sy David Friedman 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.

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.

Hugh hasn’t complained about the reduction to the Hyperuniverse, but for some reason he feels that this reduction is of no importance and has decided that all that matters is the end-product of that reduction, an analysis of ctm’s. Then he goes further and makes no distinction between the study of maximality criteria for ctm’s and the much bigger study of ctm’s in general! I really don’t understand why Hugh makes these moves.

(Fine let’s call it HP and not CTMP). HP seems now to be a one-principle program ($\textsf{SIMH}^\#$).

Further progress seems to require at the very least, understanding the implications of $\textsf{SIMH}^\#$.

As I said in my last email on this, it is impossible to have a mathematical discussion (now) of $\textsf{SIMH}^\#$ since it has been formulated so that one cannot do anything with it without first solving a number of problems which look extremely difficult. And I am not even talking about the consistency problem of $\textsf{SIMH}^\#$.

Just to be clear: This is not a criticism of $\textsf{SIMH}^\#$, it just saying it is premature to have a mathematically oriented discussion of it, and therefore of HP.

So (and this is why I have repeated myself above), I do not yet draw the distinction you do on HP versus the study of ctm’s because there not yet enough mathematical data for me to do this.

Hopefully this situation will clarify as more data becomes available. Have a productive week in California!

Concerning the future of HP, I will make a prediction: HP ends up with PD.

This has already almost happened a number of times (strong unreadability and the unreachability of V by HOD).

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 15, 2014, at 3:34 AM, Sy David Friedman wrote:

Getting as far as the $\textsf{SIMH}^\#$ is genuine progress. I provided a direction for further progress in my Maximality Protocol. Be patient, Hugh! Working out the mathematical features of Maximality will take time, and the programme has only just begun.

If things are this tentative then the conjecture “CH is false based on $\textsf{SIMH}^\#$ or some variant thereof” seems a bit curious at best.

Look we have competitor principle: $\textsf{IMH}^\#(\text{card-arith})$ which we know is consistent. There is the possibility that $\textsf{IMH}^\#(\text{card-arith})$ implies the GCH. If $\textsf{SIMH}^\#$ is inconsistent then this possibility certainly looks more likely.

As Pen has implied, it is good to have different programmes in set theory, whether they be motivated by sophisticated issues emanating from large cardinal theory and descriptive set theory, like your Ultimate-L programme, or by an “intrinsic heuristic” like the Maximality of V. Your programme is also extremely hard, but I would not fault it for that reason and I hope that it works out as hoped.

It is not whether the questions are hard which is the issue, it is whether at this stage the principles can even be discussed.

The mathematical implications of the Ultimate L Conjecture are clear and there are many. It is just the conjecture which is hard.

The mathematical implications of $\textsf{SIMH}^\#$ are not clear at all beyond failures of the GCH which are trivial. So it is somewhat difficult to have a mathematical discussion about it.

Please re-read the Maximality Protocol: Height Maximality, Cardinal Maximality, Width Maximality, in that order. I gave precise suggestions for Height and Cardinal Maximality; Width Maximality is obviously trickier but at least I made a tentative proposal with the $\textsf{SIMH}^\#$. The problem with Strong-$\textsf{SIMH}^\#$ was given toward the end of my Max story (Max isn’t happy with $\omega_1$ being captured by a single real).

So I assume you are referring to this:

The set-theorists tell him that maybe his mistake is to start talking about preserving cardinals before maximising the notion of cardinal itself. In other words, maybe he should require that $\aleph_1$ is not equal to the $\aleph_1$ of $L[x]$ for any real $x$ and more generally that for no cardinal $\kappa$ is $\kappa^+$ equal to the $kappa^+$ of $L[A]$ when $A$ is a subset of $\kappa$. In fact maybe he should go even further and require this with $L[A]$ replaced by the much bigger model $HOD_A$ of sets hereditarily-ordinal definable with the parameter $A$! [Sy's Maximality Protocol, Part 2]

Interesting. If for each uncountable cardinal $\kappa$ and for each $latex A \subset \kappa$, $(\kappa^+)^{\text{HOD}_A}$ is strictly less than $\kappa^+$ then PD holds.

About the $\textsf{SIMH}^\#$ issue. Sy, you wrote on Sept 27 in your message to Pen:

The $\textsf{SIMH}^\#$ is a “unification” of the $\textsf{SIMH}$ and the $\textsf{IMH}^\#$. The SIMH is not too hard to explain, but the $\textsf{IMH}^\#$ is much tougher. (I don’t imagine that you found my e-mail to Bob very enlightening!). Let me do the $\textsf{SIMH}$ now, and if you haven’t heard enough I’ll give the $\textsf{IMH}^\#$ a go in my next e-mail.

SIMH

The acronym denotes the Strong Inner Model Hypothesis. For the sake of clarity, however, I’ll give you a simplified version that doesn’t quite imply the original IMH; please forgive that.

A cardinal is “absolute” if it is not only definable but is definable by the same formula in all cardinal-preserving extensions (“thickenings”) of V. For example, $\aleph_1$ is absolute because it is obviously “the least uncountable cardinal” in all cardinal-preserving extensions. The same applies to $\aleph_2, aleph_3, \cdots, \aleph_\omega, \dots$ for a long way. But notice that the cardinality of the continuum could fail to be absolute, as the size of the continuum could grow in a cardinal-prserving extension (this is what Cohen did when he used forcing to make CH false; Bob Solovay got the ultimate result).

Now recall that the IMH says that if a first-order sentence without parameters holds in an outer model (“thickening”) of V then it holds in an inner model (“thinning”) of V. The SIMH says that if a first-order sentence with absolute parameters holds in a cardinal-preserving outer model of V then it holds in an inner model of V (of course with the same parameters). The SIMH implies that CH is false: By Cohen’s result there is a cardinal-prserving outer model of V in which the continuum has size at least $\aleph_2$ of V and therefore using the SIMH we conclude that there is an inner model of V in which the continuum has size at least $\aleph_2$ of V; it follows that also in V, the continuum has size at least $\aleph_2$, i.e. CH is false. In fact by the same argument, the SIMH implies that the continuum is very, very large, bigger than aleph_alpha for any ordinal alpha which is countable in Gödel’s universe L of constructible sets!

The SIMH# is the same as the SIMH except we require that V is #-generated (maximal in height) and instead of considering all cardinal-preserving outer models of V we only consider outer models of V which are #-generated (maximal in height). It is a “unification” of height maximality with a strong form of width maximality.

The attraction of the $\textsf{SIMH}^\#$ is that it is a natural criterion that mirrors both height and width maximality and solves the continuum problem (negatively).

This to me seems to clearly indicate that at that time $\textsf{SIMH}^\#$ was Strong-$\textsf{SIMH}^\#$. Sy, you wrote that “$\textsf{SIMH}^\#$ is the same as $\textsf{SIMH}$ except we require that V is #-generated…” I assumed the restriction to cardinal preserving #-generated outer models was just to simplify the discussion since otherwise $\textsf{SIMH}^\#$ would not obviously imply $\textsf{IMH}^\#$.

So fine, my assumption was not correct or you have changed. Nothing wrong with changing things, it just complicates the discussion.

In any case, perhaps it would be more efficient to postpone our discussion of HP until HP has passed the embryonic stage and things are bit more settled.

Regards.
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

So during the course of this entire email thread, HP seems to have evolved from declaring as you did in your first summary message to Sol of August 12:

I conjecture that CH is false as a consequence of my Strong Inner Model Hypothesis (i.e. Levy absoluteness with “cardinal-absolute parameters” for cardinal-preserving extensions) or one of its variants which is compatible with large cardinals existence.

(Aside: This is an extraordinary claim which you amplify in writing on Oct 13: “Further, as I said, I think there is a good chance of arriving at an optimal HP criterion”)

I stand by both of these claims, there has been no change.

to the point that it is reduced to a collection of extremely difficult problems in the structure theory of countable transitive models of ZFC; i.e. the only principle left standing seems to be $\textsf{SIMH}^\#$ (but maybe that has also fallen or been reduced to a rough uncut principle), and this has been formulated to make any near term analysis impossible. You seem to feel this is a positive attribute of HP.

Now you have lost me. You quoted a crucial phrase in my message:

Although the process exhibits genuine progress, converging closer and closer to the kind of maximality principle Max is looking for, the challenge of providing consistent forms of maximality that will answer Max’s fair questions is enormously difficult.

Getting as far as the $\textsf{SIMH}^\#$ is genuine progress. I provided a direction for further progress in my Maximality Protocol. Be patient, Hugh! Working out the mathematical features of Maximality will take time, and the programme has only just begun.

I would argue instead that this is simply a sort of coming of age for Set Theory; i.e. we can now pose simple questions about models of Set Theory which seem completely out of reach.

I am sorry that you use the phrase “which seem completely out of reach”. I would prefer to say “which demand the development of radically new techniques”. I am optimistic.

As Pen has implied, it is good to have different programmes in set theory, whether they be motivated by sophisticated issues emanating from large cardinal theory and descriptive set theory, like your Ultimate L programme, or by an “intrinsic heuristic” like the Maximality of V. Your programme is also extremely hard, but I would not fault it for that reason and I hope that it works out as hoped.

I look forward to seeing a reasoned account of HP with many of the issues that have been raised addressed. I hope that such an account has an initial list of axioms whose selection is well motivated based on the then current state of HP, i.e. why $\textsf{SIMH}^\#$ instead of Strong-$\textsf{SIMH}^\#$ etc.

Please re-read the Maximality Protocol: Height Maximality, Cardinal Maximality, Width Maximality, in that order. I gave precise suggestions for Height and Cardinal Maximality; Width Maximality is obviously trickier but at least I made a tentative proposal with the $\textsf{SIMH}^\#$. The problem with Strong-$\textsf{SIMH}^\#$ was given toward the end of my Max story (Max isn’t happy with $\omega_1$ being captured by a single real).

I’m not sure what you are asking for: To sort out this Maximality business we (and I can’t do it alone) need to explore the different possibilities and see what makes the most sense. Do you really expect me to give a precise definition of “what makes the most sense” in advance? You ask too much. It will take time, we need to map out the possibilities first.

Best of all of course would be at least one specific conjecture strongly motivated by HP ideas, whose proof confirms HP, and whose refutation is a serious setback. But I acknowledge this may not be a reasonable expectation at this preliminary stage.

I am happy to read this last sentence. What I have been trying (without much success) to say is that there is no “back to square one” conjecture here.

I have yet to see anything that suggests that HP is anything more than part of the structure theory of countable transitive models (I like Harvey’s idea; rename HP as CTMP).

??? This is very harsh. Please re-read my 13.October e-mail to Pen (the first, long one, not the second, short one). It’s about the move from Maximality features of V (as explained to Geoffrey on 24.September in terms of lengthenings, without thickenings) to the Hyperuniverse.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

On Oct 13, 2014, at 4:22 AM, Sy David Friedman  wrote:

I can only repeat what I said to you on 30.September:

Hugh, the HP is (primarily) a study of maximality criteria of the sort we have been discussing. As I have been trying to explain, it is essential to the programme to formulate, analyse, compare and synthesise different criteria, discovering their mathematical consequences. I referred to my formulation of the $\textsf{SIMH}^\#$ as “crude and uncut” as it may have to be modified later as we learn more. Changes in its formulation do not mean a defeat for the programme, but rather progress in our understanding of maximality.

and what I said to Pen yesterday:

Look, the formulations of these maximality criteria are not set in stone; if someone can suggest improvements I am happy to hear them. Indeed Hugh had some interesting suggestions in this regard. The HP is still in its infancy and there is still a lot to learn about the formulation of and the mathematics behind these criteria.

The key issue with forms of the SIMH is what to do when both of the parameters omega_1 and omega_2 are present, and not just the parameter omega_1, anyway.

Why is that the key issue?  The issue of exactly how to formulate $\textsf{SIMH}^\#(\omega_1)$ seems absolutely critical to me which is why I kept pressing you on this.

If one formulates it in terms of the preservation of $\omega_1$ then one gets a principle which strongly denies large cardinals (by implying that there is a real whose sharp does not exist).

If on the other hand, one formulates it in terms of preserving all cardinals then it does not obviously imply $\textsf{IMH}^\#$, so why the “S”?

Further formulating $\textsf{SIMH}^\#$ in terms of preserving all cardinals instead of simply adapting the formulation of $\textsf{SIMH}$ to the #-generated context, yields a principle which one cannot really do anything with current technology.  Of course one gets not-CH but this is trivial.

Let Strong-$\textsf{SIMH}^\#$ be $\textsf{SIMH}^\#$ formulated as you formulate $\textsf{SIMH}$ in your 2006 BSL paper (page 11 for those who wish to look) but for #-generated outer models. But let’s add as possible absolute parameters, definable proper classes, so that Strong-$\textsf{SIMH}^\#$ implies $\textsf{SIMH}^\#$ as you have defined it.

(Thus I am defining a proper (definable) class p to be an “class absolute parameter” if if there is a formula which defines p in all cardinal preserving outer models which are #-generated.  So the class p of all cardinals is trivially such a class. $\textsf{SIMH}^\#$ is then Strong-$\textsf{SIMH}^\#$ just restricted to class absolute parameters).

Thus Strong-$\textsf{SIMH}^\#$ implies $\textsf{IMH}^\#$ (just as SIMH implies IMH). Also Strong-$\textsf{SIMH}^\#$ implies there is a real $x$ such that $\omega_1 = \omega_1^{L[x]}$. The latter is a very deep theorem which is an immediate corollary of the sophisticated machinery of coding by class forcing that you have developed.

Maximality etc., would seem to favor Strong-$\textsf{SIMH}^\#$ over $\textsf{SIMH}^\# unless you have HP-based reasons to reject it. Do you? If not then the move to$latex \textsf{SIMH}^\# instead of to Strong-$\textsf{SIMH}^\#$ looks rather suspicious.

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Mon, 13 Oct 2014, Penelope Maddy wrote:

Dear Sy,

I didn’t change my mind about the compatibility of the $\textsf{SIMH}^\#(\omega_1)$ with all large cardinals; I just don’t see the argument anymore; Hugh, do you?

What I have been trying to say is that even if this formulation is not compatible with all large cardinals there may be other formulations which are, and we don’t know yet which formulation will prevail. That discussion is premature.

How are we to determine which formulations are worthy if we don’t pin down the statements and consequences of the candidates we have before us?

??? I already defined the IMH, SIMH and $\textsf{SIMH}^\#$ long ago. The hotly pursued $\textsf{SIMH}^\#(\omega_1)$ is just a special case of the $\textsf{SIMH}^\#$, of no special importance. It seems that Hugh focused on it because if you formulate it differently it will contradict large cardinals.

I’ll clarify things now in a message to Hugh.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I think it is great that you are getting interested in the philosophy
and foundations of set theory but you really have to do your homework
more carefully.

1. You take it as a given that the principles of Zermelo set theory
follow from (that is, are intrinsically justified on the basis of) the
iterative conception of set. This is highly contested. See, for
example, the classic paper of Charles Parsons on the topic.
2. You say that when Pen’s “Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that.”

I think you have misunderstood Pen. Thin Realism is a metaphysical thesis. It has nothing to do at all with whether the justification of an axiom references set theory alone or mathematics more generally. In fact, Pen’s Thin Realist does reference other areas of mathematics!

1. You go on to talk of three notions of truth in set theory and you
say that we should just proceed with all three. This is something that has been discussed at length in the literature of pluralism in mathematics. The point I want to make here is that it requires an argument. You cannot just say: “Let’s proceed with all three!” For
comparison imagine a similar claim with regard to number theory or physics. One can’t just help oneself to relativism. It requires an argument!

For some time now I have wanted to write more concerning your
program. But I still don’t have a grip on the X where X is your view
and at this stage I can only make claims of the form “If your view is
X then Y follows.” Moreover, as the discussion has proceeded my grip on X has actually weakened. And this applies not just to the
philosophical parts of X but also to the mathematical parts of X.

Let’s start with something where we can expect an absolutely clear and unambiguous answer: A mathematical question, namely, the question Hugh asked. Let me repeat it:

What is $\textsf{SIMH}^\#(\omega_1)$? You wrote in your message of Sept 29:

The IMH# is compatible with all large cardinals. So is the $\textsf{SIMH}^\#(\omega_1)$

It would also be useful to have an answer to the second question I
asked. The version of $\textsf{SIMH}^\#$ you specified in your next message to me
on sept 29:

The (crude, uncut) $\textsf{SIMH}^\#$ is the statement that V is #-generated and
if a sentence with absolute parameters holds in a cardinal-preserving,
#-generated outer model then it holds in an inner model. It implies a
strong failure of CH but is not known to be consistent.

does not even obviously imply $\textsf{IMH}^\#$. Perhaps you meant, the above
together with $\textsf{IMH}^\#$? Or something else?

Best,
Peter

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

So what is $\textsf{SIMH}^\#(\omega_1)$? You wrote in your message of Sept 29:

The $\textsf{IMH}^\#$ is compatible with all large cardinals. So is the  $\textsf{SIMH}^\#(\omega_1)$

A second question. The version of  $\textsf{SIMH}^\#$ you specified in your next message to me on Sept 29:

The (crude, uncut) $\textsf{SIMH}^\#$ is the statement that V is #-generated and if a sentence with absolute parameters holds in a cardinal-preserving, #-generated outer model then it holds in an inner model. It implies a strong failure of CH but is not known to be consistent.

does not even obviously imply $\textsf{IMH}^\#$.  Perhaps you meant, the above together with $\textsf{IMH}^\#$? If not then calling it $\textsf{SIMH}^\#$ is rather misleading. Either way it is closer to $\textsf{IMH}^\#(\text{card})$.

Anyway this explains my confusion, thanks.

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

Thanks, that helps. But just to be clear, does $\textsf{SIMH}^\#$ imply the  following statement?

If $\varphi$ holds of $\omega_1^V$ in a #-generated outer model of V which preserves $\omega_1^V$ then $\varphi$ holds of $\omega_1^V$ in an inner model of V.

The reason I ask is that for $\textsf{SIMH}$, the analogous statement (deleting #-generated) holds and is implied by $\textsf{SIMH}(\omega_1)$.

Regards,
Hugh