[FOM] Pluralistic Foundational Crisis?/set theory/completed

[Toby Meadows]

Dear FOMers,

I wanted to respond to some of Professor Friedman's insightful comments.
Sorry about the delay. I've just moved to Australia from the UK and I'm
still catching up with a lot of things.

I'm going to attempt to use gmail's quote-embedding to respond. Moreover,
I'll delete those parts of Friedman's comments that I'm not responding to
(in this email). This will be the third layer, but if there are any clarity
issues, I'll happily give it a tidy later.


> > Set theory is in the throes of a foundational crisis, the results of
> > which may radically alter our understanding of the infinite and
> > mathematics as a whole.
>
> It is doubtful if the specific "pluralistic crisis" being discussed
> with alter our understanding of the kind of mathematics that dominates
> mathematical practice for decades, and is expected to dominate
> mathematical practice for the foreseeable future. Working
> mathematicians have a very limited connection to set theory, using
> just enough set theory to provide a convenient underpinning for their
> real mathematical interests. When the set theory takes a life of its
> own and ceases to be convenient for their intended mathematical
> purposes, they seek to avoid it. We obviously cannot talk about
> everybody, but rather overwhelming dominating modus operandi.
>
> Nevertheless, if one adopts the viewpoint that the most, or at least
> very general concepts of set are of intrinsic importance and worth
> investigating in their own right, then one can come to be concerned
> with this "pluralistic crisis" even if it is intrinsically completely
> divorced from and totally independent from mathematical practice.
>
> In fact, there is a different kind of "foundational conundrum" in set
> theory which does promise to have a substantial and potentially
> enormous impact on mathematics as a whole. I don't think that those
> who talk about Set Theoretic Pluralism here and elsewhere have this
> different kind of "foundational conundrum" in mind, although one could
> merge the two under some wider banner. I will say a bit more about
> this later on here.
>
>
​I think Friedman's response here is quite charitable. My opening line​ is
a little over the top and I think Friedman is right to point out that
whatever crisis is brought upon set theory via multiverse interpretations,
that crisis is unlikely to have too much effect on everyday mathematical
activity.

That said, if one takes set theory seriously in a foundational sense, then
questions about how its language should be interpreted (e.g., multiverse
versus universe) seem obviously important. Moreover and depending on how
one understands the role of a mathematical foundation, the answers to such
questions could impact our understanding of mathematics as a whole, albeit
from a more philosophical (or foundational) perspective than that usually
taken up by practising mathematicians.


>
> > On
> > the second, however, significant problems have emerged. The most
> > dramatic example of this is the continuum hypothesis (CH). While the
> > large cardinal programme initially appeared to promise a means of
> > solving these kinds of problems, it is now well-known that CH is
> > independent of anything we could foreseeably think of as a large
> > cardinal assumption.
>
> It would be interesting to see a suitably general transparent notion
> of "large cardinal axiom" for which we can establish that no such can
> prove or refute CH over ZFC.
>
>
​I suppose that the closest we get here is that large, large cardinals tend
to be nicely represented by objects like ultrafilters and extenders. Using
these representations, it can then be shown that their distinctive
properties are preserved under small forcings.

This is hardly mathematical precision, although one could sharpen this kind
of thing to obtain precise definitions, which are incomplete in the sense
that they leave some large cardinals out.

I suppose the interesting question here is with regard to what kind of
problem this is. Are we seeing a failure to come up with the correct
analysis of large cardinal when there is a correct analysis to be had? Or
is the lack of a definitive analysis a kind of symptom for any programme
trying to demarcate ever higher reaches of the indefinitely extending
hierarchy of consistency strengths?



> But CH is in the family of statements whose abstract set theoretic
> intensity is way way way higher than what the working mathematician is
> focused on or feels that they really need.


​That sounds right to me in one sense and not in another.

It is surely correct that CH, as a Sigma^2_1 statement is stratospherically
out of the field of concern of working mathematicians.

But from a more "conceptual" (for want of a better word) perspective, CH
seems like a perfectly reasonable mathematical question. One can explain
Cantor's theorem to any mathematician within a few minutes. The next - and
very natural (it seems to me) - question to ask is: are there any
cardinalities strictly between that of the natural numbers and that of the
reals. It seems like a combinatorial problem.

Perhaps this "conceptual" perspective is illusory. Perhaps CH is
essentially tangled in metamathematics and set theoretic frivolity.

I also note that my talk of a "conceptual" perspective is somewhat
tangential to Friedman's remark and as such, it's not really a
counterpoint. It just seemed apropos.



> See
> http://www.cs.nyu.edu/pipermail/fom/2016-March/019584.html So even if
> the large cardinal hypotheses were to settle CH, this would not have
> been a compelling reason for the mathematical community to enlarge the
> usual ZFC foundations with large cardinal hypotheses.
>
>
​Yes, that sounds right to me. That said, it would have been - at the least
- an intriguing piece of evidence in its favour.


> > In the last few years and in response to these epistemic challenges a
> > number of new perspectives on set theory have emerged which attempt to
> > engage with these problems by avoiding the fixed ontology of the
> > cumulative hierarchy and replacing it with a plurality of universes.
> > For multiverse approaches, a problem like CH is treated as misleading
> > way of asking which universe we happen to be working in.
>
> However, it should be noted that this does not seem to apply to
> problems unlike CH in that they lie within the realm of concrete
> mathematics, of the kind that mathematicians are focused on. In
> particular, not for sentences of arithmetic or sentences of low levels
> of the analytic hierarchy.
>
>
​I think one might see Steel's version of the generic multiverse playing an
interesting part here. By considering a multiverse closed (loosely
speaking) under generic extension and refinement, we are able to see CH as
a non-concrete question. Depending on how much set theory we use to govern
each universe, we can fix more questions as being concrete. For example, if
we demand that each universe contain unboundedly many measurable cardinals,
then the Sigma^1_3 questions will be concrete in the sense that their truth
value is constant across all universes.​

This leaves open the question of how much set theory one should take up,
but from a foundational perspective, I think it is interesting to develop a
system that discards as meaningless many of those questions which are
beyond the reach of ordinary mathematics.

(I'd like to stress that this is an extremely brief and crude overview of
Steel's view but hopefully the broad strokes paint a little of the picture.)

Perhaps there are other means of achieving this effect. That would be very
interesting to me.



> > For example,
> > Joel Hamkins has proposed that set theory should be construed in
> > better faith with its practice.
>
> Perhaps foundations of set theory should be construed in better faith
> with mathematical practice?
>

​Sure.​


>
> > In accord with contemporary set
> > theory's fascination with models, Hamkins suggests that the models
> > themselves should be added its ontology (Hamkins, 2012).
>
> This reaction to different models satisfying different major set
> theoretic statements is to accept and study different models, while
> generally limiting any value judgments as to the appropriateness of
> these models.
>
> But there is another reaction to different models satisfying different
> major set theoretic statements. This is to focus on one particular
> model. A variant would be to focus on a limited group of models.
>
> We can go around in circles and say that we should pick (V,epsilon) as
> the one particular model. However, the current view based on
> experience is that this model is underdetermined. Of course, the
> diehards will in fact take the position that there is exactly one
> (V,epsilon) by definition. And therefore there is only one truth value
> of CH in that model. We just haven't yet figured out what it is.
>
> Of course, the pluralistic view is that (V,epsilon) is in fact
> underdetermined, and is not really a single model. So let's take this
> view.
>

​My anecdotally derived impression, is that the diehard (one-universe) view
is the mainstream one.​


>
> An obvious move is to focus on one particular model, and if we are
> going to focus on one particular model, the most obvious focus would
> be on (L,epsilon).
> ​​
>
> Of course (L,epsilon) has been a particularly unpopular model
> especially from those inclined to think that there is only one
> (V,epsilon). Nevertheless, it has some tremendous advantages.
>
> First and foremost, almost every single one of the known natural set
> theoretically intense statements about V(omega + omega) and many
> others, have a known truth value in (L,epsilon).
>
> So on purely mathematical set theory grounds, (L,epsilon) solves all
> of the issues, period. So why is (L,epsilon) such an unpopular choice
> of model to focus on?
>
> The usual reason given is that in (L,epsilon), there are no measurable
> cardinals (and somewhat weaker). Taking this way down in abstraction
> level, down to the relatively concrete, but still way way higher than
> the overwhelming focus of mathematical practice, (L,epsilon) satisfies
> the negation of some "good" assertions. Most notably, perhaps, is "any
> two analytic non Borel sets of reals are Borel isomorphic". This
> pleasing assertion is known to be false in (L,epsilon), but is
> provable using the existence of a measurable cardinal.
>
> I'm not convinced of the strength of this argument rejecting a focus
> on (L,epsilon). It has too much the flavor of "gee, I'll lose all this
> beautiful set theory that I've grown up with".
>

​In "Does Mathematics need New Axioms?"​ (
http://projecteuclid.org/euclid.bsl/1182353727), Steel proposes a more
nuanced argument against V=L. Essentially (and again, very briefly and
crudely), he proposes a principle of maximising interpretability. This
isn't quite a mathematically precise definition in that for a theory T to
be able to interpret (in this sense) another theory U, the interpretation
must preserve "meaning". So we might think that ZFC+\exists meas can
provide an interpretation of ZFC + V=L which preserves meaning, but we
might discard interpretations which we concoct via diagonal arguments and
Gödelian trickery as failing to preserve meaning. The definition is not
sharp.

Now Steel then argues that theories with greater interpretability (in this
restricted sense) strength are better since when we use a greater theory,
we can obtain all the results of the weaker theory by applying the
meaning-preserving interpretation. In the case of large cardinals, we might
use this principle to vaunt having a measurable cardinal above that of V=L.

There are certainly things to worry about here, but it seems to be more
principled than a mere desire to retain beautiful set theory.


>
> Another argument in favor of focusing on (L,epsilon) is that
> (L,epsilon) is the only "tangible" model containing all
> ordinals. In fact, it is the minimum model containing all ordinals, in
> the appropriate sense.
>
> It is pretty clear that no forcing extension of (L,epsilon) is going
> to be in any reasonable sense "tangible". That is, even if there are
> any. If we are in (L,epsilon) then there won't be any. Some
> interesting technical issues arise here in making clear sense of this
> (I think mostly resolved), but the main point is solid - there are no
> tangible forcing extensions of (L,epsilon). One can only look to
> studying families, abandoning the very idea of focusing on one model.
>
> ADDED 4/9/16: However, there are tangible Boolean valued models.
> Boolean valued models are not "official" models. I am still not clear
> how or whether Boolean valued models enters the philosophical picture.
> One view is that they are perfectly legitimate models as much as
> official models are. Another view is that they are simply a very
> convenient way of talking about a big family of non tangible models.
>
> Granted there is a level of tangibility around in models like
> (L(mu),epsilon), where mu is a kappa additive measure on a measurable
> cardinal kappa. But still there is the question of where that kappa
> and where that measure comes from. How do we determine whether a
> subset of kappa is to have measure 0 or have measure 1? It is my
> impression that the attempts to deal with this issue are not
> satisfactory. You seem to have to say that some involved process
> actually miraculously works to get this.
>

​I'd like to get a better idea of the notion of "tangibility" at play here.

Rather than consider L(\mu), what about L[0#]. Would that count as
tangible? What about the hierarchy of models that then lead us from L[0#]
upward to core models?​


>
> We an also go further and consider the notion of ordinal as
> underdetermined just as we have considered the notion of V as
> underdetermined. Then we arrive at the so called minimum transitive
> model, which is a countable (L(lambda),epsilon). Enough of this
> discussion...
>
> > John Steel
> > takes the impressive impact of the large cardinal programme on
> > descriptive set theory and turns our ordinary understanding of sets on
> > its head. Rather than thinking of set theory as describing some
> > pre-existing structure in which mathematics can be seen to take place,
> > we should rather see it as a congenial scaffolding through which
> > further concrete mathematics can be interpreted (Steel, 2012).
>
> This use of the word "concrete" must be distinguished from normal
> mathematical usage. I would say that the Borel measurable world (in
> Polish spaces) is at the outer limits of anything that would even
> remotely be regarded as concrete from the point of view of
> mathematical practice. But I need to take a careful look at Steel,
> 2012 to further address Steel's viewpoint.
>
>
​Yes, I think the use of the word "concrete" is somewhat vexed, but I'm not
sure this is merely a definition debate. I think what is - at least in part
- at stake in something like Steel's programme is an effort to expand the
ordinary mathematical usage of the term. ​


> > Finally, Friedman’s hyperuniverse programme attempts to combine
> > features of both the universe and multiverse perspectives. By tracking
> > first order properties of universes in multiverses constrained by
> > natural principles, Friedman aims to discover new axiom candidates to
> > characterise the universe of sets V.
>
> There was a very extensive discussion of S. Friedman's hyperuniverse
> program on a blog some time ago, and I think that this blog has been
> retained for retrieval by Koellner.
>
> The program was at least originally billed as generated by a new
> analysis of the idea of V being "maximal". However, S. Friedman's
> approach to this in terms of inner models is very brittle in the sense
> that everywhere you turn, a slight change bites you in the face with
> an inconsistency.
>
> To address this brittleness, I proposed that this idea of "maximality
> of V" be revisited carefully with philosophical coherence already with
> PA, Z2, Z, to ZFC, looking for new ideas beyond the existing
> literature on maximality in set theory.
>
> At the time the ideas in this HP program looked much more like a
> systematic study of countable models of ZFC, a perfectly respectable
> technical program in set theory, like Hamkin's. At least that was the
> then assessment of both Woodin and me.
>
> Also the jump from features of countable models of ZFC to what is or
> should be true in V had a lot of difficulties at the time.
>
>
​I don't have anything to add here. The extension discussion from last year
is well worth reading.​



> > Väänänen uses his dependence
> > logic, in particular the concept of team semantics, to make sense of
> > the multiverse idea. His starting point is general first order logic
> > with multiverse structures and he applies this to set theory.
>
> I am not familiar with this approach, and would like to take a look.
>

​Here is a link:
http://www.math.helsinki.fi/logic/people/jouko.vaananen/multiverse5.pdf.​


> >
> > Each of these pictures admits a kind of pluralistic ontology and
> > indeterminacy into foundations. The move is controversial. Hugh Woodin
> > has argued that the kind of generic multiverse offered by Steel
> > reduces set theory to a species of formalism that betrays its
> > Cantorian roots (Woodin, 2012).
>
> Yes, this conforms to my impression that Woodin, along with his thesis
> advisor Solovay, are among the really prominent diehards arguing for
> absolute reality and matters of fact for (V,epsilon).
>
> > Moreover, Tony Martin has offered a
> > naïve re-working of Zermelo's categoricity argument to claim that the
> > indeterminacy revealed by CH is of a merely epistemic nature and thus,
> > that the metaphysical re-imaginings of Hamkins and Steel are
> > unwarranted (Martin, 2001; Zermelo, 1976). In a related vein, a
> > criticism of the pluralist account of foundations is given by Väänänen
> > in his comparison of the second order logic and set theory approaches
> > (Väänänen, 2012).
>
> Martin has said a long time ago that the longer we go without any
> convincing assignment of a truth value to CH, the weaker the case for
> naive realism, let alone Platonic reality. But Martin thinks that the
> advances in set theory since he said this makes him more rather than
> less optimistic. I don't share that assessment of what has transpired.
> >
> > Beyond the mathematical challenges involved in addressing these
> > programmes, there are significant overlaps with recent work in
> > mainstream analytic philosophy, particularly in metaphysics and
> > philosophical logic.
>
> My general impression has been that work in metaphysics and
> philosophical logic would not be able to come to grips with the rather
> focused issues in foundations of set theory.
>
> Actually I hope that my general impression is wrong! This would open
> up a rather exciting and possibly productive adventure.
>

​I'd like to be more optimistic here. I think there are a number of younger
philosophers who possess the requisite skills and interest for this
adventure. Time will tell!​


>
> However, there is one thing that I strongly believe. In order to get
> the philosophical side of things to really engage with the foundations
> of set theory, there has to be a very much upgraded line of
> communication between people proving deep theorems and people doing
> intricate philosophy. In order for this to happen, both sides will
> have to rethink their entire approaches from first principles and
> engage with each other.
>
> I cannot tell from the ensuing paragraphs to what extent this
> interactive soul searching is taking place.
>
>
​I think this is right. Personally, this why I wanted to set up this
network. I'd like to see more philosophers and mathematicians just talking
to each other. Indeed, I should note that this network was largely inspired
by a conference in London set up by Sean Walsh in 2011. It was
well-subscribed by mathematicians and philosophers and, I think, a lot of
​productive relationships came out of it.



> > A key problem in metaontology is Putnam’s
> > paradox, which is a generalisation of Skolem’s paradox to language and
> > semantics at large.
>
> Already right here, there is probably a kind of clash of cultures.
> Math logicians have been trained to think that there is no paradox of
> any kind in Skolem.
>
>
​I tend to the no-paradox view myself, however, I cited this topic more as
a location where philosophers can benefit from the insights of mathematical
logicians. For example, the starting point for this argument occurs in
Putnam's "Models and Reality" (
https://www.princeton.edu/~hhalvors/teaching/phi520_f2012/putnam1980.pdf)​,
which contains a relatively technical argument (by philosophical standards)
about L and omega-models. I should also stress that Putnam's paradox, while
related to Skolem's paradox, is a little more subtle and difficult to
escape.



> > Using model theoretic techniques, it is argued
> > that we are caught in a regress of theory augmentation whenever we
> > seek to give a full account of the meaning of our expressions.
>
> Meaning that if we have a theory, and we want to reflect on it, and
> treat the meaning of the expressions used, we must add to the theory,
> a la Tarski undefinability of truth, etc.
>
>
​It's not so much a Tarskian hierarchy in the sense that the metalinguistic
jumps are forced by a diagonal argument. Rather, we try to provide an
extensional theory for the meaning of the terms of our language. Then using
what is known as the permutation argument, we find that those terms could
actually denote different things. So we expand our theory in attempt to
clarify this. Then the permutation argument can be applied again ... and so
on.

To be honest, I'm not doing this any justice. A good source for this is
David Lewis's "Putna's​ Paradox" (
http://web.mit.edu/rvm/www/metaphysics%20reading%20group/Lewis%20-%20Putnam's%20Paradox.pdf
).



> > Without
> > such an account, we lose the ability to anchor our ontology to our
> > language.
>
> I half understand this sentence, but would be interested in having it
> explained.
>
>
I think I'll demur and direct you back to the Lewis paper. ​



> > A response emerges with Lewis and has been developed by
> > Sider, Schaffer and Williams. They argue that there is a privileged
> > language which carves nature at it joins and that this is the goal of
> > our best theories.
>
> I would like to see this in action, so I get a feel for what "carves
> nature at it joins" means. And examples of "privileged languages".
>
>
​I sit on other (more Carnapian, less platonism) side of the fence to these
views, but I can point out a couple of good places to start.

   - David Lewis's "Putnam's Paradox" again! but toward the end.​ (
   http://web.mit.edu/rvm/www/metaphysics%20reading%20group/Lewis%20-%20Putnam's%20Paradox.pdf
   )
   - Ted Sider's "Writing the Book of the World" (sample -
   http://tedsider.org/books/wbotw_sample.pdf)

​I think it's fair to say that these kinds of views are big business in
mainstream analytic philosophy's metaphysical quarters. ​



> > For multiverse debates, these approaches are
> > particularly useful for the one-universe adherent.
>
> I'm curious what the one-universe adherents like Woodin and Solovay
> and Koellner do with this.
>
>
​Perhaps I should have given this remark more thought. My inclination is to
think of "joint carving" theories as being - more or less (much discussion
required) - platonist in close to the Gödelian sense of that term.

I've also been tempted to ​think that belief in a one-universe requires
some kind of intense realism somewhere in the vicinity of platonism. So one
might argue that there is one universe on the basis of the axioms of ZFC
carving at the joins of nature in the sense of articulating truths about
"the" membership relation.

That said, there is certainly some wiggle room here. I don't see why the
multiverse adherant couldn't be a platonist (Hamkins claims to be one); and
I don't see why the universe adherent needs to be a platonist: perhaps the
one-universe stance is ultimately more congenial. Nonetheless, I still get
the feeling that realism sits more comfortably with the one-universe view;
perhaps this is just a prejudice.



> > Related work by
> > Kennedy (2013) suggests a pluralistic approach involving generalised
> > constructibility and more widely the concept of "formalism freeness",
> > and its dual, the concept of the entanglement of a semantically given
> > object with its underlying formalism.
>
> I am a little bit familiar with generalized constructibility, and I
> assume that "formalism freeness" means a deliberate non commitment to
> any fixed formalism? I would like an elaboration of what
> "entanglement" means here, e.g., by examples.
>
>
​I might invite Professor Kennedy to talk about this.​



> > On the other hand, there has
> > also been recent work into the identification of substantive debates.
> > Stemming from Carnap (1956) and Ryle (1954) – and emerging more
> > recently with Thomason (2009), Chalmers (2011) and Sider (2011), it is
> > argued that some metaphysical debates are merely verbal. Such debates
> > are pointless as although the parties to the debate are in conflict
> > nothing substantive hangs on the result. With multiverse debates,
> > these approaches provide a means of arguing that some questions are
> > meaningless.
>
> Yes, there really is the crucial question of whether phrases like
> "matters of fact", "objective reality", "one or multi universes", are
> meaningful.
>

​This is an area where I believe philosophers can add a lot to the debate.
This is a very general issue affecting a lot of disciplines and it has
received a lot of philosophical attention in recent ​years. The sources
cited above are very good. Moreover, they attempt to provide criteria for
assessing the substantive-ness of questions.

​I also believe that the sharpness of mathematical examples could help
further understanding of these problems in philosophy. ​


> More broadly, we can demand that philosophical debates be
> "productive". But then, what kind of "production" are we looking for,
> or should be looking for?
>
> Speaking for myself, I like to make the distinction between
> foundations and philosophy. E.g., foundations of mathematics is NOT
> the same as philosophy of mathematics, although there are a lot of
> interactions and common interests.
>
>
​I think this is important, but I'd propose looking at it from a slightly
different perspective.

I'd like to suggest that foundations of mathematics fits straddles both
philosophy and mathematics (mostly mathematical logic).

However, on the philosophical side, I see there are some reasons for
reservation. Much activity in philosophy of mathematics has focussed on
questions of metaphysics and epistemology. So philosophers have asked
questions like: what is a number? and how can we know about the numbers?
Such questions are certainly of core philosophical concern, however, I
don't think they are of much interest to working mathematicians, nor are
they particularly relevant to the foundations of mathematics.

I think this could give one reason to think foundations aren't part of
philosophy of mathematics. But I think this is misleading. It seems to me
that beyond the technical "internal" questions (involving the solving of
mathematical problems etc), the kinds of "external" questions about which
framework is good, right, etc are clearly philosophical ... at least they
don't look like the kinds of questions working mathematicians tend to ask.
 ​
​Perhaps this will degenerate into a definition debate.​



> For me, "productive" means whether it furthers foundational research,
> and I have some sort of working "definition" for what foundational
> research is, and what ideally comes out of it.
>
> In general, I have not found that philosophers generally make the
> moves that further foundational research. However, I do find that when
> I talk to philosophers one on one, and try to explain foundational
> research to them, things come up that I haven't thought of before, and
> this often opens up new lines of foundational research.
>
>
​Perhaps you could give an example of a move that furthers foundational
research. I suppose I suspect that I'll agree that philosophers aren't
making those moves, but perhaps there are other concerns that philosophers
can (are?) contributing to which have foundational value. I don't know;
perhaps this will be an epicycle on the definition of foundations. ​


> > With regard to philosophical logic, a significant amount of recent
> > activity has been devoted to problems of indeterminacy; in particular,
> > problems caused by vagueness and the liar paradox.
>
> Of course, the whole thrust of mathematics is to avoid vagueness and
> paradoxes. And of course, vagueness and paradoxes are firmly embedded
> in ordinary language.
>
>
​Agreed, although the techniques for dealing with vagueness are frequently
applicable to other paradoxes and situations involving indeterminacy.


> > A prominent
> > response to these problems is known as supervaluation. Observing that
> > indeterminacy results where there are different possibilities none of
> > which is determined as correct, supervaluation tells us that the
> > determinate propositions are those which are true regardless of which
> > possibility we select.
>
> Of course, the great classic thingie like this is Goedel's
> Completeness Theorem. I recall that this has been tried with languages
> that support directly and indirectly, self reference as an attempt to
> deal with Liar Paradox and related paradoxes. But I never found what
> came out of this very attractive. There is absolutely no comparison
> between the fixes of the Russell Paradox and the fixes of the Liar
> Paradox. Actually, I have had it on my wish list to fix the Liar
> Paradox with the same level of clarity and robustness as has been done
> with the Russell Paradox. I think this can be done, but seems
> difficult.
>
>
​I'd like to hear more about this!​


> > In the context of the multiverse, a proposition
> > is meaningful if it is true in every universe.
>
> Do you mean to say "a proposition is meaningful if it has the same
> truth value in every universe", or "a proposition is meaningful if and
> only if it has the same truth value in every universe"?
>
>
​That was sloppy of me. I should have written the latter of your
formulations. ​


> > one of many different approaches to indeterminacy which include
> > epistemicism, fuzzy logic, non-standard consequence relations and
> > paraconsistency (Williamson 2008).
>
> I have not seen any of these things interact decently with foundations
> of set theory - at least not yet.
>
>
​I'm hoping that something along at least one of these lines will emerge
from our network.​


> > It has been observed that any
> > approach to indeterminacy developed in one area can be generalised
> > into an analogous response in another. This raises interesting
> > questions about the applicability of a wider variety of techniques in
> > philosophical logic to the multiverse.
>
> I would like to see what "analogous response" means here.
>
>
​Here's what I had in mind. When considering the Sorites paradox, one can
consider a number of "solutions" including: strong Kleene semantics;
supervalution; epistemicism and the logic of paradox (dialeithism).

Each of these approaches has a relatively natural counterpart "solution"
for the liar paradox. One may question just how natural the counterpart
relation is here.

My (wildly speculative) thought here was that multiverse solutions tend
provide a supervaluation approach to indeterminacy. They discard CH since
it is true in some valuations/worlds and false in others. One might wonder
if other approaches to indeterminacy (from philosophical logic) could also
be used to deal with set theoretic indeterminacy. One might ask: to what
end? Well, perhaps a different perspective could make the indeterminacy
easier to explain an understand (a predominantly philosophical goal).
Regardless, the idea was a bit of a lob into open territory. Perhaps
someone out there will have the requisite skills and background to see how
to do this easily or, indeed, why it is a non-starter.



> Harvey Friedman
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-- 
Toby Meadows
Lecturer, Department of Philosophy
School of Divinity, History and Philosophy
University of Aberdeen · Old Brewery · High Street · Aberdeen · AB24 3UB
https://sites.google.com/site/tobymeadows/
https://www.facebook.com/PhilUniAb

Set Theoretic Pluralism: https://sites.google.com/site/pluralset/home

*I am a physical object sitting in a physical world. Some of the forces of
this physical world impinge upon my surface. Light rays strike my retinas;
molecules bombard my eardrums and fingertips. I strike back, emanating
concentric air waves. These take the form of a torrent of discourse about
tables, people, molecules, light rays, retinas, air waves, prime numbers,
infinite classes, joy and sorrow, good and evil. [Quine]*
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