Re: Paper and slides on indefiniteness of CH

Dear Hugh,

$\textsf{IMH}^\#(\text{card-arith})$ is consistent relative to a bit more than one Woodin cardinal, in fact it holds if every real has a sharp and there is a countable ordinal which is a Woodin cardinal in an inner model. But the proof requires the adaptation of Jensen coding to $\#$-generated models etc.  There is a natural guess as to the statement of that theorem and I am assuming it holds: If M is an inner model of GCH which is $\#$-generated then there is a real $x$ such that M is a definable inner model of $L[x]$, $x^\#$ exists, and such that $L[x]$ is a cardinal preserving extension of M.

Yes, you get this using my proof with Radek of the consistency of $\textsf{IMH}^\#$, observing that with GCH presrvation, the proof goes through without collapsing any cardinals. Actually you get more: The inner model is also $\#$-generated. The key to all of this is Jensen’s proof (see Chapter 9 of the coding book) that one has coding with $\#$-generation.

Actually I think that you need what Radek and I use (coding with $\#$-generation) even to handle the following consistent very special case of $\textsf{SIMH}^\#$: If a sentence with parameter $\omega_1$ holds in a cardinal-preserving $\#$-generated outer model then it holds in a $\#$-generated inner model with the same $\omega_1$.

As for $\textsf{IMH}(\text{card-arith})$, a natural conjecture is that $\textsf{IMH}^\#(\text{card-arith})$ implies GCH.

Maybe it is “natural” but I don’t see the evidence for it. In my view this conjecture implies that the SIMH is inconsistent. On the other hand I must confess that I don’t see the evidence for the consistency of the SIMH either!

I have no idea about the consistency of $\textsf{IMH}^\#(\text{card})$ which looks like the problem of showing $\textsf{SIMH}$ is consistent.

I would guess that the situation with $\textsf{IMH}^\#(\text{card})$ is not very different than the situation with $\textsf{IMH}(\text{card})$, which is very much like the SIMH.

Now a variant of a problem which I raised much earlier in  this email-thread seems quite relevant. The new problem is: Can there exist a $\#$-generated M such that if N is a $\#$-generated extension which is cardinal preserving then every set of N is set-generic over M.

I think this is an interesting question without $\#$-generation; I’m not sure what $\#$-generation adds to it. I think that a negative answer would follow if one could do what I did with forcing with finite conditions shooting clubs through $\omega_2$, but with $\omega_2$ replaced by $\text{Ord}$ and also preserving GCH. Krueger and Mota recently verified my conjecture that one can do the $\omega_2$ case preserving GCH.

Sy

Re: Paper and slides on indefiniteness of CH

Dear all,

I believe this concludes the discussion that Sy and I were having on $\textsf{IMH}^\#$.  I would like to note one more thing and here the proof does likely requires the machinery Sy was discussing.

Let $\textsf{IMH}^\#(\text{card})$ be $\textsf{IMH}^\#$ together with if there is a cardinal preserving $\#$-generated extension in which $\varphi$ holds then there is a cardinal preserving inner model in which $\varphi$ holds. Similarly define $\textsf{IMH}^\#(\text{card-arith})$ (see earlier in the thread) by restricting to $\#$-generated extensions and inner models, which preserve all cardinals and cardinal arithmetic.

$\textsf{IMH}^\#(\text{card-arith})$ is consistent relative to a bit more than one Woodin cardinal, in fact it holds if every real has a sharp and there is a countable ordinal which is a Woodin cardinal in an inner model. But the proof requires the adaptation of Jensen coding to $\#$-generated models etc.  There is a natural guess as to the statement of that theorem and I am assuming it holds: If M is an inner model of GCH which is $\#$-generated then there is a real $x$ such that M is a definable inner model of $L[x]$, $x^\#$ exists, and such that $L[x]$ is a cardinal preserving extension of M.

As for $\textsf{IMH}(\text{card-arith})$, a natural conjecture is that $\textsf{IMH}^\#(\text{card-arith})$ implies GCH.

I have no idea about the consistency of $\textsf{IMH}^\#(\text{card})$ which looks like the problem of showing $\textsf{SIMH}$ is consistent.

Now a variant of a problem which I raised much earlier in this email-thread seems quite relevant. The new problem is: Can there exist a $\#$-generated M such that if N is a $\#$-generated extension which is cardinal preserving then every set of N is set-generic over M.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

Sorry, I forgot to clarify the situation with the IMH variants that you raised. It seems that you are interested in “imaginable extensions” which are thickenings (outer models).

On Sat, 20 Sep 2014, Harvey Friedman wrote:

I would like to focus the discussion on the proper formulation of the IMH. The formulations being emphasized by Sy do not seem to be satisfactory from a foundational perspective, and their justifications offered by Sy have been sharply criticized here by the philosophers.

Nevertheless, there is an interesting idea here and goes like this.

THESIS 1. Any first order sentence that holds in some imaginable extension of V holds in some part of V.

Implicit in my IMH.

THESIS 2. Any imaginable extension of V satisfies the same first order sentences as some part of V.

Inconsistent.

These suggest the following strengthenings:

THESIS 3. Any second order sentence that holds in some imaginable extension of V holds in some part of V.

If you mean to work with MK models, then this is a minor variant of my IMH, using a result of mine with Carolin Antos (on coding MK models into reals).

THESIS 4. Any imaginable extension of V satisfies the same second order sentences as some part of V.

Inconsistent.

There seem to be several main ways to formulate these theses, with various outcomes and advantages and disadvantages. But this suggests

THESIS 5. Any first order sentence that holds in some imaginable binary relation on the ordinals holds in some actual binary relation on the ordinals.

Consistent.

1. Any first order sentence (set of first order sentences) forced to
hold by a set forcing over V holds in some transitive class containing
all ordinals.

2. Any first order sentence (set of first order sentences) forced to
hold by a class forcing over V holds in some transitive class
containing all ordinals.

1 and 2 are weak versions of my IMH, except if you say “set of first order sentences” then they become inconsistent.

3. There is a transitive class M containing all ordinals, satisfying ZFC, such that every first order sentence (set of first order sentences) that holds in some such M’, holds in some such M” contained in M.

Inconsistent.

4. There is a transitive class M containing all ordinals, satisfying
ZFC, such that every first order sentence (set of first order
sentences) that holds in some such M” contained in some such M’, holds in some such M”’ contained in M.

Same as 3.

5. There is a countable transitive model M of ZFC such that every first order sentence that holds in some such containing M holds in some such included in M.

Version of the IMH.

6. There is a countable transitive model M of ZFC such that every such containing M satisfies the same first order sentences as some such contained in M.

Inconsistent.

VERSION 1

There is the infinitary language $L_{\text{Ord},\omega}$ with ordinary quantifiers but arbitrary On length conjunctions and disjunctions. There are well known axioms and rules of inference. This gives us a notion of set theoretic consistency, where we put down obvious axioms – ZFC, every set has a rank less than $\omega$ (formulated with lots of large conjunctions and disjunctions), and ask that adding a given first order (finitely) sentence phi does not throw the system into inconsistency. Then we assert that if so, then there is a countable transitive model containing all ordinals of $\varphi$.

As I said before, this is inconsistent.

VERSION 2.

Same as version 1. Except this time, we throw in the appropriate diagram for V to “ensure that imaginary models will extend actual V”.

Then you have my version of the IMH formulated in “V-logic”.

In summary: Your versions are either already implicit in my IMH or are inconsistent.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Pen and others,

On Sat, 20 Sep 2014, Penelope Maddy wrote:

Dear Sy,

Speaking just for this one non-mathematician, I’d very much appreciate being allowed to continue following this discussion.

Sorry! I didn’t think that you (and others) would be interested in such a technical discussion. Here is what you missed: Honzik and I showed that there is a real $R$ such that any $\#$-generated universe containing $R$ satisfies the $\textsf{IMH}^\#$, the “IMH for $\#$-generated universes” and that the $\textsf{IMH}^\#$ is compatible with arbitrary large cardinals. But we used some determinacy to prove that and Hugh claimed that the result is “trivial” using just sharps for reals. I knew that the result was indeed derivable from $\#$‘s for reals, but thought that the proof was far from “trivial”. Hugh convinced me otherwise; below is his argument for those who are interested. $R$ is chosen to be a real such that $R^\#$ exists and $R$ computes an element of any nonempty lightface $\Pi^1_2$ set.

Let $M$ be a $\#$-generated model to which $R$ belongs. Let $M^*$ be an extension of $M$ which is $\#$-generated and in which there is a definable inner model $M_0$ with  $M_0 \vDash \varphi$. Let $(N,U)$ witness that $M^*$ is $\#$-generated. Then

$N_{\kappa} \vDash \text{There is a definable inner model }M_0 \vDash \varphi"$

where $\kappa$ is the largest cardinal of $N$ (since $N_\kappa$ is an elementary substructure of $M^*$).

The set of all reals $x$ which code a countable iterable $(N,U)$ for which $N_{\kappa} \vDash \text{There is a definable inner model }M_0 \vDash \varphi"$ where $\kappa$ is the largest cardinal of $N$, is a lightface-$\Pi^1_2$-set. So there is a such an iterable model $(N,U) \in M$. This gives a definable inner model of $M$ in which $\varphi$ holds.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

I would like to focus the discussion on the proper formulation of the IMH. The formulations being emphasized by Sy do not seem to be satisfactory from a foundational perspective, and their justifications offered by Sy have been sharply criticized here by the philosophers.

Nevertheless, there is an interesting idea here and goes like this.

THESIS 1. Any first order sentence that holds in some imaginable extension of V holds in some part of V.

THESIS 2. Any imaginable extension of V satisfies the same first order sentences as some part of V.

These suggest the following strengthenings:

THESIS 3. Any second order sentence that holds in some imaginable extension of V holds in some part of V.

THESIS 4. Any imaginable extension of V satisfies the same second order sentences as some part of V.

There seem to be several main ways to formulate these theses, with various outcomes and advantages and disadvantages. But this suggests

THESIS 5. Any first order sentence that holds in some imaginable binary relation on the ordinals holds in some actual binary relation on the ordinals.

Now it is my impression that Thesis 5 formulated in various ways is inconsistent (I may be confusing this with something else). Then we have to ask what went wrong, as thesis 5 does seem to address the idea of the maximality of proper classes. And what is wrong with maximality of proper classes? We need to get clear about just what is OK and what is not. My general impression is that things at present do not have the robustness than we want for a fundamental theory – at least not yet.

E.g., maybe there is a very clear “nonmaximality theorem for proper classes”???

I will consider only formulations in the very convenient MKGC = Morse Kelly with the global axiom of choice. The formulations involving countable transitive models are completely straightforwardly formulated in ZFC (or very weak fragments thereof) and there are no issues of formulation.

1. Any first order sentence (set of first order sentences) forced to hold by a set forcing over V holds in some transitive class containing all ordinals.
2. Any first order sentence (set of first order sentences) forced to hold by a class forcing over V holds in some transitive class containing all ordinals.
3. There is a transitive class M containing all ordinals, satisfying ZFC, such that every first order sentence (set of first order sentences) that holds in some such M’, holds in some such M” contained in M.
4. There is a transitive class M containing all ordinals, satisfying ZFC, such that every first order sentence (set of first order sentences) that holds in some such M” contained in some such M’, holds in some such M”’ contained in M.
5. There is a countable transitive model M of ZFC such that every first order sentence that holds in some such containing M holds in some such included in M.
6. There is a countable transitive model M of ZFC such that every such containing M satisfies the same first order sentences as some such contained in M.

In contrast to the formulations 1-4, my understanding is that 6 is inconsistent. So when we move to countable transitive models, bad things happen that don’t happen otherwise. To me, this makes it clear that we cannot have any confidence that the structure of countable transitive models of ZFC give us good information about V.

Sy has been emphasizing 5, and at least implicitly also using 3 or 4. 2 is mentioned in passing in Sy, Phillip, Hugh paper.

My favorite formulation, which has two obvious versions, uses infinitary languages. This is an attempt to have a principled idea of “imaginary universe”. Both are formulated in MKGC.

VERSION 1

There is the infinitary language $L_{\text{Ord},\omega}$ with ordinary quantifiers but arbitrary $Ord$ length conjunctions and disjunctions. There are well known axioms and rules of inference. This gives us a notion of set theoretic consistency, where we put down obvious axioms – ZFC, every set has a rank less than $\omega$ (formulated with lots of large conjunctions and disjunctions), and ask that adding a given first order (finitary) sentence $\varphi$ does not throw the system into inconsistency. Then we assert that if so, then there is a countable transitive model containing all ordinals of $\varphi$.

VERSION 2.

Same as version 1. Except this time, we throw in the appropriate diagram for V to “ensure that imaginary models will extend actual V”.

It is my understanding that version 1 is inconsistent, and that version 2 is a good formulation of IMH.

But this is another warning sign of non robtusness. Version 1 should be OK, at least for class maximality.

I’ll stop here. I hope that the usual suspects on this list will clarify the exact mathematical situation, and the other usual suspects on this list will clarify the philosophical issues.

Harvey

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Fri, 19 Sep 2014, W Hugh Woodin wrote:

Dear Sy,

I looked at your paper with Honzik which discusses IMH# and which you referred us to in the outline on HP which you circulated. I am rather confused about the discussion of IMH#. It seems like the conclusion of Thm 4.2 and Thm 4.3 of the paper both follow trivially from just sharps.

This is not the case for 2 reasons. First, you need to perform Jensen coding for $\#$-generated universes, a major refinement of the ordinary Jensen coding. Second, to convert the existence of a definable inner model satisfying some sentence into a $\Sigma^1_3$ property requires a version of what I call “David’s Trick”, another significant refinement of Jensen coding. And you then have to combine these two refinements.

For these reasons I chose in the paper to make things easy and use Determinacy. The proof can however be done with much more work using just a real with a sharp that computes an element of each nonempty lightface $\Pi^1_2$ set.

Hugh, this is a very technical discussion, not appropriate for the entire group. If you want to continue it, please don’t bother everybody but just write to me (and perhaps a few other mathematicians who would genuinely be interested).

Regards,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I looked at your paper with Honzik which discusses $\textsf{IMH}^\#$ and which you referred us to in the outline on HP which you circulated.

I am rather confused about the discussion of $\textsf{IMH}^\#$.  It seems like the conclusion of Thm 4.2 and Thm 4.3 of the paper both follow trivially from just sharps. Let $R$ be real such that every nonempty lightface-$\Sigma^1_3$ set contains a member recursive in $R$. Then any M which is $\#$-generated and contains $R$, satisfies $\textsf{IMH}^\#$.

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

On Sun, 14 Sep 2014, Harvey Friedman wrote:

I just sent this posting to the FOM email list:

I am participating in a small email group concerning higher set theory, focusing originally on whether the continuum hypothesis is a “genuine” problem. The discussion has partly gone into issues surrounding the foundations of higher set theory. The ideas in this posting were inspired by the interchange there.

CAUTION: I am not an active expert in this kind of higher set theory, and so what I say may be either known, partly true, or even false.

As a modification of existing ideas concerning “maximality” I offer the following axiom over MK class theory with the global axiom of choice.

DEFINITION 1. Let $\varphi$ be a sentence of set theory. We say that phi is set theoretically consistent if and only if ZFC + $\varphi$ is consistent with the usual axioms and rules of infinitary logic. At the minimum, the standard axioms and rules of $L_{V,\omega}$, with quantifiers ranging over $V$, with $\in,=$, but one may consider the much stronger language $L_{V,\text{Ord}}$ where set length blocks of quantifiers are used. I think that for what I am going to do, it makes no difference, so that there is stability here.

Then we can formulate in class theory,

POSTULATE A. If a sentence is set theoretically consistent then it holds in some transitive model of ZFC containing all ordinals. It follows that it holds in some $L[x]$, $x$ a real.

As formulated, both parts of this are inconsistent. The first part becomes a weak version of the IMH if you require consistency with the correct base theory (diagram of V plus every ordinal is an ordinal of V). The second statement gives an inconsistency no matter what you do.

For a better understanding of the connection between outer models and consistency in infinitary logic, I recommend Mack Stanley’s paper. I used this connection to formulate the IMH in a weak 2nd order set theory (roughly speaking, it is 1st order over the “least admissible set past V”).

Now what happens if we relativize this in an obvious way?

DEFINITION 2. Let $\alpha$ be an infinite ordinal. Let $\varphi$ be a sentence of set theory. We say that $\varphi$ is set theoretically consistent relative to $P(\alpha)$ if and only if ZFC + $\varphi$ is consistent with the usual axioms and rules of infinitary logic together with “all subsets of $\alpha$ are among the actual subsets of $\alpha$“, where the latter is formulated in the obvious way using infinitary logic.

POSTULATE $\text{B}_\alpha$. If a sentence is set theoretically consistent relative to $P(\alpha)$ then it holds in some transitive model of ZFC containing all ordinals.

POSTULATE C. Postulate $\text{B}_\alpha$ holds for all infinite $\alpha$.

To prove consistency, or consistency of this for some $\alpha$, we seem to need

1. An extension of Jensen’s coding the universe where we add a subset of $\alpha^+$ that codes the universe without adding any subsets of $\alpha$. This has probably been done.

No, this is not possible if CH fails.

1. Cone determinacy for subsets of $\alpha^+$.E.g., using the equivalence relation $x \sim y$ if and only if $x,y$ are interdefinable over $V_{\alpha^+}$. I don’t know if this has been done.

This is inconsistent.

You seem to be trying to generalise the IMH in various ways. See the Strong IMH in my BSL paper and the variants mentioned in the HP outline that I sent on 13.September.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

This note touches on three matters.

1. A correction to the SRM = strict reverse math system I presented.

The error and its correction are quite interesting. I’m writing a new paper on SRM which will have the right statement and break new SRM ground.

2. Formulations of the IMH = inner model hypothesis.

3. Concerning my need for a response to the Fields Medalist as to “why should I believe a subtle cardinal exists or is consistent?”

I wrote

1. Linearly ordered integral domain axioms.
2. Finite interval. $[x,y]$ exists.
3. Boolean difference. $A\setminus B$ exists.
4. Set addition. $A + B = \{x+y: x \in A\text{ and }y \in B\}$ exists.
5. Set multiplication. $A\times B = \{xy: x \in A\text{ and }y \in B\}$ exists.
6. Least element. Every nonempty set has a least element.
7. $n^0 = 1$. $m \geq 0$ implies $n^{m+1} = n^m \times n$. $n^m$ is defined only if $m \geq 0$.
8. $\{n^0,\dots,n^m\}$ exists.

and said that the above is a conservative extension of EFA = exponential function arithmetic = $\textsf{I}\Sigma_0(\text{Exp})$.

8 needs to be changed to

8. $\{0+n^0,1+n^1,...,m+n^m\}$ exists.

This issue is not a problem in SRM formulations with finite sequences — we can get away with postulating that $(n^0,n^1,...,n^m)$. In any case, one first proves that 1-7 above is conservative over PFA = polynomial function arithmetic = $\textsf{I}\Sigma_0(\text{Exp})$.

“I am sorry that you take such a negative view of this programme’s philosophical merits. But at least you may have interest in the math that comes out of it!”

It appears that to some extent, Sy has followed a model of thought that I indicated in an earlier email. Namely, he felt some Foundational Traction with IMH, and then developed and fine tuned an associated philosophy. This is, for various reasons, evidently not received positively clearly by Pen and Peter, and to some extent, not by Hugh and John either.

I can see some Foundational Traction in the IMH. However, it needs to be reformulated in order to have more. I quote from “On the consistency strength of the Inner Model Hypothesis“:

The Inner Model Hypothesis (IMH): If a statement $\varphi$ without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V.

Equivalently: If a statement $\varphi$ is internally consistent in some outer model of $V$ then it is already internally consistent in $V$. This is formalised as follows. Regard $V$ as a model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that $V^*$ is another such model, with the same ordinals as $V$. Then $V^*$ is an outer model of $V$ ($V$ is an inner model of $V^*$) iff the sets of $V^*$ include the sets of $V$ and the classes of $V^*$ include the classes of $V$. $V^*$ is compatible with $V$ iff $V$ and $V^*$ have a common outer model.

CAUTION: I AM NOT AN ACTIVE EXPERT IN THIS KIND OF HIGHER SET THEORY I AM A SEMI-CASUAL CONSUMER. SO WHAT I SAY BELOW MAY BE KNOWN, OR ONLY PARTIALLY TRUE, OR COMPLETELY FALSE.

They show that, using some PD (and hence a big enuf large cardinals by Martin/Steel), for all reals $x$ of sufficiently high Turing degree, the least transitive model of ZFC containing $x$ satisfies IMH. They also use IMH to get models of strong kinds of measurable cardinals using inner model theory for strong kinds of measurable cardinals (Mitchell). There seems to be good deep higher set theory here.

If I were writing that paper, I would have included some friendly warmup material to allow casual and semi-casual readers to get more out of it. In particular, a warmup proof that $0^\#$ comes out of IMH, presumably using Jensen’s original covering theorem. And maybe a second warmup for $0^\dagger$. Also, there may be strategic warmups for the construction of models of IMH. But maybe there are some expositional notes on IMH that are more friendly??

The formulation is somewhat awkward for any kind of intrinsic treatment of set theoretic maximality. Clearly IMH has something to do with maximality, but the philosophy gets strained under the present formulation of IMH (at least the formulation from this IMH paper).

IMH as a statement in class theory paradoxically is formulated in terms of outer models of V. But according to maximality, V doesn’t have any (proper) outer models. So the outer models of V must be taken to be imaginary.

So one way to go is this:

INFORMAL HYPOTHESIS. If a sentence holds in some imaginary transitive model of ZFC containing all ordinals, then it holds in some actual transitive model of ZFC containing all ordinals.

An obvious candidate for “imaginary transitive model of ZFC” is given by class forcing, where of course one does not get involved with any actual generic objects – because V is in fact maximal and so generic objects don’t really exist. I.e., on this interpretation, a sentence holds in an “imaginary transitive model of ZFC” if and only if it is forced by all conditions in some appropriate class forcing that does exist. Appropriate means that it is ZFC friendly.

I guess that this is equivalent by Jensen coding to INFORMAL HYPOTHESIS. There is a real number $x$ such that the following holds. If a sentence holds in some imaginary transitive model of ZFC containing all ordinals, then it holds in some inner model of $L[x]$.

Now class forcing seems to be too technical to be a satisfactory interpretation of “imaginary transitive model of ZFC”. Instead, we can simply require that a sentence be consistent with a strong form of the axioms and rules of inference of infinitary logic applied to ZFC. This seems more general than class forcing. Using definable cone determinacy and Jensen coding, it does appear that the infinitary proof rule formulation can be shown to be consistent – by throwing it into countable models, where the infinitary axioms and rules of inference are complete and hence give actual models to do Jensen coding on.

Thus on this formulation, countable models only appear in a consistency proof, and not in the actual formulations.

With regard to my question about what the best answer is to that senior Fields Medalist who asks “Why is there a subtle cardinal, or why is it consistent with ZFC?” There is the following exchange so far on the FOM.

Rupert McCallum writes:

William Tait wrote an essay that appeared in “The Provenance of Pure Reason” called “Constructing Cardinals from Below” which discussed a set of reflection principles that justify SRP. Unfortunately Peter Koellner later observed that some of the reflection principles he considered were inconsistent. I wrote down my own thoughts in a recent Mathematical Logic Quarterly article about how one might find principled grounds for distinguishing the consistent ones from the inconsistent ones.

I sent in to FOM:

I’m sure that the FOM readers would be most interested if you could give a simple brief account of the ideas behind some of the reflection principles that work – at least in the sense that they can be obtained from standard large cardinal hypotheses. Of course, subtle cardinals themselves are based on a very simple idea – but that idea would not normally be characterized as reflection.

For just subtle, we have $\kappa$ is essentially subtle if and only if $\kappa$ is a cardinal such that for all binary relations $R$ on $kappa$, there exists infinite $\alpha < \beta < \kappa$ such that the sections of $R$ at $\alpha,beta$ agree below $\alpha$.

Note that essentially subtle is closed upward, so it is not quite the same as being subtle. HOWEVER, the first subtle cardinal is exactly the first essentially subtle cardinal. ALSO “there exists a subtle cardinal” is equivalent to “there exists an essentially subtle cardinal”.

If FOM readers relate to your simple brief account, they can of course delve into publications. FOM readers can also get a chance to interact online starting from what you write.

Harvey

PS: Maybe I see how to do this using some arguable reflection using multiple universes. Let’s consider two universes $V$ and $V'$, where $V'$ is longer than $V$. Let’s not worry about the most philosophically honest way to formalize this just yet.

Let $R$ be a binary relation on $V$ and let $\varphi$ be a sentence that holds in $(V',V,R)$. “Reflection” says that there exists kappa in V such that $\varphi$ holds in $(V',V_kappa, R|V_kappa)$. This seems to prove Con(ZFC + “there exists a subtle cardinal”). I think that if you use $V,V',V'',V''',\dots$ then you will get Con(SRP).

Re: Paper and slides on indefiniteness of CH

Dear Peter,

Thanks for your message, especially the first half with helpful clarifications regarding the proper use of the terms “extrinsic” and “intrinsic”. I especially like your suggestion of considering “degrees of intrinsicness” (“relativised” versions) and this should fit very well with the HP. This is perhaps anticipated by Pen’s comment near the end of her 27.August message, where she reports that I might consider “returning to the proposal of a different conception of set. The challenge there is to do so without returning to the unappealing idea that ‘intrinsic justification’ and ‘set-theoretic truth’ are determined by a conception of the set-theoretic universe that’s special to a select group.” One could perhaps interpret “special to a select group” as “low degree of intrinsicness” in your sense.

But more to the point, Peter, the situation is as follows: Pen and I initiated a process of carefully examining the steps in the HP and got stuck. As Pen said in her 27.August message, one of the key steps is my claim to extract something new from the Maximal Iterative Conception (MIC). Her message includes a description of my claim, which includes the use of “lengthenings” and “thickenings” of mental pictures of V. Pen felt that there was “something off about a universe being ‘maximal in width’, but also having a ‘thickening'” and I replied that “lengthenings” were already implicit in reflection (Pen’s message includes my argument for that). At that point Pen decided it would be best to consult with you directly about the role of “lengthenings” in reflection and explicitly asked for your opinion. What do you think? I’d really appreciate your response because Pen and I have been waiting for it since August 27!

Regarding the second half of your message: I do not claim that the IMH is intrinsically justified! This is partly my fault, since my views have changed since Tatiana and I wrote our paper. In my 7.August message to Pen I say:

… as my views have evolved slightly since Tatiana and I wrote the BSL paper I’d like to take the liberty (see below) of fine-tuning and enhancing the picture you present above. My apologies for these modifications, but I understand that changes in one’s point of view are not prohibited in philosophy?

I go on to say:

… what I came to realise is that the IMH deals only with “powerset maximality” and it is compelling to also introduce “ordinal maximality” into the picture. (I should have come to
that conclusion earlier, as indeed the existence of inaccessible cardinals is derivable from the intrinsic maximal iterative concept of set!)

And further on:

This was an important lesson for me and strongly confirms what you suggested: In the HP (Hyperuniverse Programme) we are not able to declare ultimate and unrevisable truths. Instead it is a dynamic process of exploration of the different ways of instantiating intrinsic features of universes, learning their consequences and synthesising criteria together with the long-term goal of converging towards a stable notion ….

I do understand that I said very different things in my original paper with Tatiana, but I tried to correct this in my 7.August e-mail to Pen. And I do realise that it is too much to ask that you sift through all of those e-mails I sent to Pen (there were many!) so I’m happy to repeat things now to sort out misunderstandings.

In abridged form: The HP starts with intrinisic features of V that follow from the Maximum Iterative Conception and then provides a method for turning these features into precise mathematical criteria which ultimately yield first-order statements. The process is dynamic, whereby the choice of criteria together with their first-order consequences can change over time, as indicated in the last quote above. So one does not arrive at “unrevisable intrinsically justified” statements, as sometimes criteria and their consequences are discarded as the programm progresses. This already happened to the IMH: it must be synthesised with ordinal-maximality, and if this is done using the Friedman-Honzik form of reflection ($\#$-generation) this removes its anti-large cardinal consequences.

The above description obviously leaves huge gaps, in particular it does not explain what the mathematical criteria are about and how they are chosen. But it gives the rough idea and I hope clarifies that the word “intrinsic” is intended to apply to features of V and only to criteria and their consequences after a lengthy (practice-independent) process of analysis and synthesis has occurred. I plan to examine each feature of the HP carefully in further discussions with Pen. But we are in need of your response to her message!

Thanks,
Sy

PS: The HP is not concerned with justifications of consistency. The consistency of large cardinals is taken as given in the programme.