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

[Harvey Friedman]

This is the completion of my previous posting Pluralistic Foundational
Crisis?/set theory at
http://www.cs.nyu.edu/pipermail/fom/2016-April/019652.html

For the reader's convenience, this posting incorporates the previous
http://www.cs.nyu.edu/pipermail/fom/2016-April/019652.html so that
this posting is entirely self contained.

The in depth scientific announcement is rather detailed and is at
https://sites.google.com/site/pluralset/context-aims-of-the-network

There is a lot here that I think is very appropriate for elaborations
on the FOM. I would think that the various "pluralistic communities",
including philosophers, could benefit from having their ideas
discussed here on the FOM.

********************

> 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.

> Over the course of the twentieth century, set theory has become the de
> facto foundation for mathematics. It plays this role in two ways.
> First, it provides an ontology of spaces and objects which can
> represent the subject matter of contemporary mathematics. Second, it
> provides a lever via which the problems of contemporary mathematics
> may be solved. On the first front, set theory has been a success.

This is not controversial.

> 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.

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. 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.

> 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.

> 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?

> 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.

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".

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.

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.

> 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.

> 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.
>
> 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.

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.

> 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.

> 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.

> 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.

> 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".

> 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.

> 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.

> 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.

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.

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.

> 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.

> 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.

> 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"?

> 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.

> 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.

Harvey Friedman