# Re: Paper and slides on indefiniteness of CH

Dear Pen and Sy,

I have benefited from your exchange. I’ll try to add some input.

Sy: I have wanted to say something about your proposal. But I am still very unclear on how you understand the philosophical landscape, in particular, on how you understand the distinction between intrinsic and extrinsic justification.

One of the distinctive aspects of your view — a selling point, from a philosophical point of view — is that it involves pursuit of “intrinsic justifications”, as opposed to “extrinsic justifications”. But I am not sure how you are using these terms. From your exchange with Pen it seems that your usage is quite different from the original, namely, that of Gödel.

For Gödel a statements S is intrinsically justified relative to a concept C (like the concept of set) if it “follows from” (or it “unfolds”, or is “part of”) that concept. The precise concept intended is far from clear but it seems clear that whatever it is intrinsic justifications are supposed to be very secure, not easily open to revision, and qualify as analytic. In contrast, on your usage it appears that intrinsic justifications need not be secure, are easily open to revision, and (so) are (probably) not analytic.

For Gödel a statement S is “extrinsically justified” relative to a concept C (like the concept of set) if it is justified (on the basis of reasons grounded in that concept) in terms of its consequences (especially its “verifiable” consequences), just as in physics. Again this is far from precise but it seems clear that extrinsic justifications are not as secure as intrinsic justifications but instead offer “probable”, defeasible evidence. In contrast, on your usage it appears that you do not understand “extrinsic justification” as an epistemic notion, but rather you understand it as a practical notion, one having to do with meeting the aims of a pre-established practice.

So, you appear to use “intrinsic justification” for an epistemic notion that is not as secure as the traditional notion but rather merely gives epistemic weight that falls short of being conclusive. Moreover, at points, when talking about intrinsic justifications you talk of testing them in terms of their consequences. So I think that by “intrinsically justified” you mean either “intrinsically plausible” or “extrinsically justified”.

I think you need to be more precise about how you use these terms and how your usage relates to the standard usage. This is especially important if the main philosophical selling point of your proposal is that it is re-invigorating “intrinsic justifications” in the sense of Gödel. (Good places to start in getting clear on these notions are the papers of Tony Martin and Charles Parsons.)

In what I say next I will use “intrinsic justification” in the standard sense, both for the sake of definiteness and because it is on this understanding that your view is distinctive from a philosophical point of view.

Let me begin with a qualification. I am generally wary of appeals to
“intrinsic justification”, for the same reason I am generally wary of
appeals to “self-evidence”, the reason being that in each case the
notion is too absolute — it pretends to be a final certificate, an
epistemic high-ground, a final court of appeal. But in fact there is
little agreement on what is intrinsically justified (and on what is
self-evident). For this reason, in the end, discussions that employ
these notions tend to degenerate into foot-stamping. It is much
better, I think, to employ notions where there is widespread
intersubjective agreement, such as the relativized versions of these notions, notions like “A is more intrinsically plausible than B” and “A is more (intrinsically) evident than B”. This is one reason I find
extrinsic justifications to be more helpful. They are piecemeal and
modest and open to revision under systematic investigation. (I think
you agree, since I think that ultimately by “intrinsic justification”
you mean what is normally meant by “extrinsic justification”).

But let me set that qualification aside and proceed, employing the notion of “intrinsic justification” in the standard sense, for the
reasons given above.

There is an initial puzzle that arises with your view.

THE PUZZLE

1. You claim that IMH is intrinsically justified.
2. You claim that inaccessible cardinals — and much more — are intrinsically justified
3. FACT: IMH is implies there are no inaccessibles.

The natural reaction at this point would be to think that there is

But perhaps there is a subtlety. Perhaps in (1) and (2) intrinsic
justifications are relative to different conceptions.

When you claim that IMH is intrinsically justified what exactly are you saying and what is the case for the claim? Are you saying IMH (a) intrinsically justified relative to our concept of set (which, on the face of it, concerns V) or (b) the concept of being a countable transitive model of ZFC, or (c) the concept of being a countable transitive model of ZFC that meets certain other constraints? Let’s go through these options one by one.

(a) IMH is intrinsically justified relative to the concept of set. I don’t see the basis for this claim. To the extent that I have a grasp on the notion of being intrinsically justified relative to the concept of set I can go along with the claims that Extensionality and Foundation are so justified and even the claims that Infinity and Replacement and Inaccessibles are so justified (thus following Gödel and others) but I lose grip when it comes to IMH. Moreover, IMH implies that there are no inaccessibles. Does that not undermine the claim that IMH is intrinsically justified on the basis of the concept of set? Assuming it does (and that this is not what you claim) let’s move on.

(b) IMH is intrinsically justified relative to the concept of being a
countable transitive model of ZFC. I have a good grasp on the notion of being a countable transitive model of ZFC. And I think it is interesting to study this space. But when I reflection this space –when I try to unfold the content implicit in this idea — I can reach nothing like IMH.

(c) IMH is intrinsically justified relative to the concept of being a countable transitive model of ZFC that meets certain other constraints. I can certainly see going along with this. But, of course, it depends on what the other constraints are. We have two options: (i) We can be precise about what we mean. For example, we can build into the notion that we are talking about the concept of being a countable transitive model of ZFC that satisfies X, where X is
something precise. We might then deduce IMH from X. In this case we know what we are talking about — that is, we know the subject matter — but we merely “get out as much as we put in”. Not so interesting. (ii) We can be vague about what we mean: For example, we can say that we are talking about countable transitive models of ZFC that are “maximal” (with respect to something). But in that case we have little idea of what we are talking about (our subject matter) and it seems that “anything goes”.

You seem to want to resolve the conflict in (a) — between the claim
that inaccessibles are intrinsically justified and the claim that IMH
is intrinsically justified — by resorting to both intrinsic justifications on the basis of our concept of set (which gives inaccessibles) and intrinsic justifications on the basis of the hyperuniverse (understood as either (i) or (ii) under (c)) and which gives IMH) and you seem to want to leverage the interplay between these two in such a way that it gives us information about our concept of set (which concerns V). But what can you say about the relationship between these two forms of intrinsic justification? Is there some kind of “meta” (or “cross-domain”) form of intrinsic justification that is supposed to give us confidence about why intrinsic justifications on the basis of the hyperuniverse should be accurate indicators of truth (or intrinsic justifications on the basis of) our concept of set?

One final comment: Here is an “intuition pump” regarding the claim that IMH is intrinsically justified.

THE FACTS:

1. If there is a Woodin cardinal with an inaccessible above then IMH
is consistent.
2. If IMH holds then measurable cardinals are consistent.

So, if IMH is intrinsically justified (in the standard sense) then we can lean on it to ground our confidence in the consistency of measurable cardinals. For my part, the epistemic grounding runs the other way: IMH provides me with no confidence in the consistency of measurable cardinals (or of anything). Instead, the consistency of IMH is something in need of grounding. Fact (1) above provides me with evidence that IMH is consistent. Fact (2) does not provide me with evidence that measurable cardinals are consistent. I think most would agree. If I am correct about this then it raises further problems for the claim that IMH is intrinsically justified (in the standard sense).

I have further comments and questions about your notion of “sharp-generated reflection” and how you use it to modify IMH to $\textsf{IMH}^\#$. But those questions seem premature at this point, given that I am not on board with the basics. Let me just say this: The fact that you are readily modify (intrinsically justified) IMH to $\textsf{IMH}^\#$ in light of the fact that IMH is incompatible with (intrinsically justified) inaccessibles indicates that your notion of intrinsic justification is quite revisable and, I think, best regarded as “intrinsic plausibility” or “extrinsic justification” or something else

Best,
Peter

# Re: Paper and slides on indefiniteness of CH

Hugh Woodin wrote to Sy Friedman:

I will try one more time. At some point HP must identify and validate a new axiom. Otherwise HP is not a program to find new “axioms”. It is simply part of the study of the structure of countable wellfounded models no matter what the motivation of HP is.

It seems that to date HP has not done this. Suppose though that HP does eventually isolate and declare as true some new axiom. I would like to see clarified how one envisions this happens and what the force of that declaration is. For example, is the declaration simply conditioned on a better axiom not subsequently being identified which refutes it? This seems to me what you indicate in your message to Pen.

Out of LC comes the declaration “PD is true”. The force of this declaration is extreme, within LC only the inconsistency of PD can reverse it.”

As I understand it, the original IMH put forward by Sy has the following drawbacks. Both of you were involved in getting upper and lower bounds on the strength of original IMH.

1. The official presentation is in terms of countable transitive models, and not a sentence or scheme in set theory itself. Thus in a careful presentation of IMH, one defines what a countable IMH model is, and the IMH states that there is a countable IMH model.
2. Sy proves that no countable IMH model satisfies that there exists an inaccessible cardinal.
3. There are versions of IMH, not the ones emphasized, that are statements in class theory – incidentally, not set theory – involving class forcing. Then one can talk about a theory extending NBG + Global Choice. And this theory, as I understand it, proves that there are no inaccessible cardinals.
4. Of course, the existence of an inaccessible cardinal represents (a consequence of) a much clearer and convincing kind of maximality than IMH with proper class forcing. Countable model IMH – the official version – doesn’t represent any kind of maximality principle whatsoever involving sets and classes. So to turn this around, inaccessible cardinals refute IMH (class forcing).
5. As I understand it, there has been a reworking of IMH so that the new form(s) are known to be, or conjectured to be, consistent with some large cardinals. In order to address the first sentence above from Hugh Woodin, these new form(s) need to be formulated not as modified IMH (countable models) but as modified IMH (class forcing), in order for people to examine what their value is for “intrinsic” foundations of set theory. I understand that, technically, there may be little or no difference between the two formulations of modified IMH (at least in terms of results), but there is a major difference in terms of direct relevance to the “intrinsic”.
6. I glanced briefly at some account of one of these variants of IMH – yes for countable models, but presumably it has a class forcing version — and it was at least one layer more involved than the original IMH (any form). My concern is that accommodating the IMH idea with large cardinals will look like an ad hoc glueing of separate ideas, but what I briefly glanced at was not friendly reading.
7. Of course, you can simply start with ZFC + LCs, and look at only “outer models” satisfying ZFC + the same LCs. This would be the most obvious plan. Does anything interesting come out of this simple minded fix (countable models or class forcing forms)?
8. What would you two recommend in terms of the clearest statement of “second generation” IMH that accommodates at least some large cardinals so that we can judge how natural this fix (or multiple fixes) of IMH is?
9. I have no criticism per se of people treating countable transitive models of ZFC in a way roughly akin to, say, group theorists treating countable groups (although of course there are very substantial connections with other areas of mathematics, which we do not have for ctm’s) – as specialized mathematical activity. What is at issue is whether such investigations rise to the level of affecting our understanding of the foundations of set theory – or whether they hold any promise for doing so.

Any comments from any of you two should be illuminating.

Harvey

# Re: Paper and slides on indefiniteness of CH

Dear Sol and others,

On Mon, 11 Aug 2014, Solomon Feferman wrote:

I wrote that “[CH] can be considered as a definite logical problem relative to any specific axiomatic system or model.  But one cannot say that it is a definite logical problem in some absolute sense unless the systems or models in question have been singled out in some canonical way.”

And with regard to Hugh’s proposed Ultimate-L and my proposed SIMH, Sol wrote:

In both these cases, if the proposed “solution” fails, CH is left in limbo.

Yes, but is asserting that CH is a definite logical problem the same as solving it? Perhaps the HP could in time be recognised as a framework in which the “models in question have been singled out in some canonical way”, and if so, the definiteness of CH as a logical problem is thereby established, even without providing the solution. This is what I was getting at in my August 7 mail to you, where I presented the SIMH.

One thing is clear: If Hugh succeeds mathematically with Ultimate-L and I succeed mathematically with the SIMH, then CH is again “left in limbo”! It will then come down to the legitimacy of the two corresponding and different approaches to truth, Hugh’s “practice-based” approach and my “intrinsically-based” approach. That’s one reason I felt it of key importance to carefully examine the underlying assumptions of the HP with Pen, as if the grogramme is flawed then its conclusions can’t be trusted. Moreover I never intended the HP to be just about CH and I never intended the SIMH to be the only way to use the programme to settle CH.

Now I come to the main purpose of this mail. I said some very strong things in my mails to Pen, and Pen has responded clearly and thoroughly to them.  In my view, Pen’s reduction of “truth” to set-theoretic practice qualifies her as a true “radical”, in the best sense of the word! So I think that I can fairly assume that not all of you philosophers out there agree with Pen and I would like to hear your reactions to some of the strong assertions I have made:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

…what is the point of trying to clarify truth in set theory? I never imagined that it was to guide the future development of set theory!

… it is precisely because … the subject is constantly changing that I don’t want to base my theory of truth on set-theoretic practice. I am looking for ‘hard core intrinsically-based truth’, even if it conflicts with what is suggested by current practice … This is not quite as radical as it may seem. We accept the axioms of ZFC as being true, on intrinsic grounds. But in set-theoretic practice we are constantly working with models where powerset, replacement or choice fails. So accepting the axioms of ZFC as being true has not ‘jettisoned’ good mathematics carried out in settings where those axioms fail.

… My ‘gripe’ is when good mathematics gets promoted to the status of ‘discovery about truth’ without adequate justification … I would like philosophers to take a more active role in preventing this from happening. If there is something that I would like to discard it is not good mathematics but the misuse of good mathematics to make unjustified claims about set-theoretic truth.

You may feel that … we should welcome any investigation which a mathematician reassures us is relevant to the investigation of truth. But surely if the conclusions of such an investigation are interesting, such as a solution to CH, we would want to verify that the arguments which led there were well-grounded philosophically and that there were not mathematical choices made along the way just to make things work.

I can tell you as a mathematician that it is not hard to deceive oneself into thinking that one’s exciting new results have important implications for truth in set theory … And aren’t I being currently subjected to a valuable ‘grilling’ by an expert in the philosophy of mathematics (you)? I think that any mathematician who claims to investigate truth should be subjected to such a ‘grilling’. Philosophers of mathematics: We need you!

Best regards to all,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Penny,

On Wed, 6 Aug 2014, Penelope Maddy wrote:

As I now (mis?)understand your picture, it goes roughly like this … We reject any ‘external’ truth to which we must be faithful, but we also deny that the resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).  One key is that ‘true-in-V’ is answerable, not to a realist ontology or some sort of ‘truth value realism’, but to various intrinsic considerations.  The other key is that it’s also answerable to a certain restricted portion of the practice, the de facto set-theoretic claims.  These are the ones that ‘due to the role that they play in the practice of set theory and, more generally, of mathematics, should not be contradicted by any further candidate for a set-theoretic statement that may be regarded as ultimate and unrevisable’ (p. 80).  (Is it really essential that these statements be ‘ultimate and unrevisable’?  Isn’t it enough that they’re the ones we accept for now, reserving the right to adjust our thinking as we learn more?)  These include ZFC and the consistency of LCs. The intrinsic constraints aren’t limited to items that are ‘implicit in the concept of set’.  They also include items ‘implicit in the concept of a set-theoretic universe’.  (This sounds reminiscent of Tony’s reading in ‘Gödel’s conceptual realism’.  Do you find this congenial?)  One of the items present in the latter concept is a notion of maximality.  The new intrinsic considerations arise at this point, when we begin to consider, not just V, but a range of different ‘pictures of V’ and their interrelations in the hyperuniverse.  When we do this, we come to see that the vague principle of maximality derived from the concept of a set-theoretic universe can be made more precise — hence the schema of Logical Maximality and its various instances. At this point, we have the de facto part of practice and various maximality principles (and more, but let’s stick with this example for now).  If the principles conflict with the de facto part, they’re rejected.  Of the survivors, they’re further tested by their ability to settle independent questions. Is this at least a bit closer to the story you want to tell?

Yes, but 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?

As you say, I take “true in V” to be free of any realist ontology: there is no fixed class of objects constituting the elements of the universe of all sets. But this does not prevent us from having a conception of this universe or from making assertions about what is true in it. My notion of set-theoretic truth (truth in V) consists of those conclusions we can draw based upon intrinsic features of the relevant concepts. The relevant concepts include of course the concept of “set”, but also (and this is a special aspect of the HP) the concept of “set-theoretic universe” (“picture of V”).

Intrinsic features of the concept of set include (and in my view are limited to) what one can derive from the maximal iterative concept (together with some other basic features of sets), resulting in the axioms of ZFC together with reflection principles. These are concerned with “internal” features of V.

To understand intrinsic features of the concept of set-theoretic universe we need a context in which we may compare universes and this is provided by the hyperuniverse. The hyperuniverse admits only countable universes (countable pictures of V) but by Löwenheim-Skolem this will suffice, as our aim is to clarify the truth of first-order statements about V. An example of an intrinsic feature of the concept of universe is its “maximality”. This is already expressed by “internal” features of a universe based on the maximal iterative concept. But in the HP it is also expressed by “external” features of a universe based on its relationship with other universes (“maximal” = “as large as possible” and the hyperuniverse provides a meaning to the term “possible universe”).

With this setup we can then instantiate “maximality”, for example, in various ways as a precise mathematical criterion phrased in terms of the “logic of the hyperuniverse”. The IMH (powerset maximality) and $\#$-generation (ordinal maximality) are examples, but there are others which strengthen these or synthesise two or more criteria together (IMH for $\#$-generated universes for example). The “preferred universes” for a given criterion are those which obey it and first-order statements that hold in all such preferred universes become candidates for axioms of set theory.

With this procedure we have a way of arriving at axiom-candidates that are based on intrinsic features of the concepts of set and set-theoretic universe. A point worth making is that our notions of V and hyperuniverse are interconnected; neither is burdened by an ontology yet they are inseparable, as the hyperuniverse is defined with reference to V and our understanding of truth in V is influenced by the (intrinsically-based) preferences we impose on elements of the hyperuniverse.

What has changed in my perspective since the BSL paper (I cannot speak for Tatiana) regards the “ultimate” nature of what the programme reveals about truth and the relationship between the programme and set-theoretic practice. Penny, you are perfectly right to ask:

Is it really essential that these statements be ‘ultimate and unrevisable’?  Isn’t it enough that they’re the ones we accept for now, reserving the right to adjust our thinking as we learn more?

At the time we wrote the paper we were thinking almost exclusively of the IMH, which contradicts the existence of inaccessible cardinals. This is of course a shocking outcome of a reasoned procedure based on the concept of “maximality”! This caused us to rethink the role of large cardinals in set-theoretic practice and to support the conclusion that in fact the importance of large cardinals in set theoretic practice derives from their existence in inner models, not in V. Indeed, I still support that conclusion and on that basis Tatiana and I were prepared to declare the first-order consequences of the IMH as being ultimate truths.

But 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!) There are various ways to formalise ordinal maximality as a mathematical criterion: If we take the line that Peter Koellner has advocated then we arrive at something I’ll call KM (for Koellner maximality) which roughly speaking asserts the existence of omega-Erdos cardinals. A much stronger form due to Honzik and myself is $\#$-generation, which roughly speaking asserts the existence of any large cardinal notion compatible with V = L. Now IMH + KM is inconsistent but we can “synthesise” IMH with KM to create a new criterion IMH(KM), which is consistent. Similarly we can consistently formulate the synthesis IMH($\#$-generation) of IMH with $\#$-generation. Unfortunately IMH(KP) does not change much, as it yields the inconsistency of large cardinals just past $\omega$-Erdos, and so again we contradict large cardinal existence. But the surprise is that IMH($\#$-generation) is a synthesised form of powerset maximality with ordinal maximality which is compatible with all large cardinals (even supercompacts!), and one can argue that #-generation is the “correct” mathematical formulation of ordinal maximality.

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 of “preferred universe”. At each stage in the process, the first-order statements which hold in the preferred universes can be regarded as legitimate axiom candidates, providing an approximation to “ultimate and unrevisable truth” which may be modified as new ideas arise in the formulation of mathematical criteria for preferred universes. Indeed the situation is even more complex, as in the course of the programme we may wish to consider other intrinsic features of universes (I have ignored “omniscience” in this discussion), giving rise to a new set of mathematical criteria to be considered. And it is of course too early to claim that the process really will converge towards a unique notion of “preferred universe” and not to more than one such notion (fortunately there are as of yet no signs of such a bifurcation as “synthesis” appears to be a very powerful and successful way of combining criteria).

Finally: Why do I refer to “axiom candidates” and not to “axioms” when I mention first-order properties shared by preferred universes? This is out of respect for “set-theoretic practice”. As you know my aim is to base truth wholly on intrinsic considerations, independent of what may be the current trends in the mathematics of set theory. In my BSL paper we try to fix a concept of defacto truth and set the ground rule that such truth cannot be violated. My view now is rather different. I see that the HP is the correct source for axiom candidates which must then be tested against current set-theoretic practice. There is no naturalist leaning here, as I am in no way allowing set-theoretic practice to influence the choice of axiom-candidates; I am only allowing a certain veto power by the mathematical community. The ideal situation is if an (intrinsically-based) axiom candidate is also evidenced by set-theoretic practice; then a strong case can be made for its truth.

But I am very close to dropping this last “veto power” idea in favour of the following (which I already mentioned to Sol in an earlier mail): Perhaps we should accept the fact that set-theoretic truth and set-theoretic practice are quite independent of each other and not worry when we see conflicts between them. Maybe the existence of measurable cardinals is not “true” but set theory can proceed perfectly well without taking this into consideration. In the converse direction I simply repeat what I said recently to Hugh:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

The HP is about intrinsic sources of truth and we have no a priori guarantee that the results of the programme will fit well with current set-theoretic practice. What to do about that is however unclear to me at the moment.

All the best, thanks again for your interest,
Sy