[FOM] CFR and Programme: SoTFoM III and The Hyperuniverse Programme, Vienna, 21-23 September 2015.

[Harvey Friedman]

I am planning to implement an idea aimed at expanding the scope of the
FOM discussions.

The idea is simple. When I see a talk announcement title, or a talk
abstract, or a manuscript, that at least promises to have relevance to
the foundations of mathematics, I will call it to the attention of the
FOM and make a brief (or not so brief) comment about it, and invite
the author(s) to post on it, from the f.o.m. perspective.

One rich source, but by no means the only source, of such items can be
found in the now many meetings organized with strong mathematical
logic and philosophy of mathematics presence.

Inevitably, there will be many items that I don't mention, that could
have great relevance to f.o.m. I strongly urge FOM readers to consider
doing at least some of the same, or at least call my attention
privately to items that I have missed.

I want to start this idea now. Several of the titles of the talks
announced in the subject header meeting fit into this category. I have
not seen abstracts to these talks. When will they be available?

Looking at the program, these titles are the ones that suggest f.o.m.
relevance. Of course, I am only going by the titles, and would be
better informed by the abstracts.

Giorgio Venturi: `Forcing, Multiverse and Realism'

Daniel Waxman and Jared Warren: `Is there a good argument for
mathematical pluralism?'

Øystein Linnebo: `Potentialism about set theory.'

Geoffrey Hellman: `A Height-Potentialist Multiverse View of Set Theory.'

Sam Sanders: `Non-standard analysis as a computational foundation.'

Emil Weydert: `A multiverse axiom induction framework.'

Douglas Blue: `Forcing axioms and maximality as the demand for
interpretability.'

Peter Koellner: `On the Multiverse Conception of Set.'

I INVITE ALL OF THE ABOVE AUTHORS TO POST ON THE FOM GIVING AT LEAST A
SYNOPSIS OF THEIR MAIN POINTS AND FINDINGS THAT ARE RELEVANT TO F.O.M.
Posting only links without discussion can be very valuable, but the
inclusion of generally understandable elaborations is generally much
more helpful.

In the meantime, I make some comments.

GENERAL COMMENTS

Many set theorists, including the mature Goedel, adopted a strongly
Platonist view of mathematics, and particularly set theory. That the
objects have an objective existence that is very much like that
normally attributed to physical objects. At least sharing the supposed
property that all statements about them have a definite truth value. A
number of difficulties have emerged over decades with regard to this
kind of Platonism, although the strongest adherents deny that any
difficulties have emerged. More nuanced views have emerged, in
particular various forms of Realism, led by Naive Realism, which does
not adhere to objective existence of objects, but does adhere to
definite truth values of all assertions. And there has been some
difficulties in properly distinguishing between these two views.
Originally, the idea has been that there is only one fundamental
concept of set, period. All other notions are derived. The very idea
of multiple universes was an anathema. But with certain developments
and especially certain non developments, Among these are the notorious
failure to arrive at a truth value for practically all of the set
theoretic statements known to be independent of the ZFC axioms,
through intellectual reflection on the unique set theoretic universe.
However, controversially, some of these statements have, to various
degrees, been objectively assigned a truth value this way, or at least
through some sort of naive realism. In any case, there is general
agreement that for certain statements of historical mathematical
interest, namely statements concerning arbitrary sets of real numbers
(surrounding CH, etc.), reflection on the unique set theoretic
universe has been a failure. Two main attempts at salvaging the
situation have emerged. First there is the idea that there is a kind
of notion of experimental evidence that is somehow akin to that used
in physical science, with the idea of confirmation and falsifiability.
the second main attempt is a multiple universe idea. Both attempts
have major difficulties.With respect to the first, the notions of
confirmation and even much more so, the notion of falsifiability, seem
very unclear and unsatisfying particularly in comparison to
confirmation and falsifiability in physical  science. With respect to
the second, there is the problem of just what alternative universes
one wants to admit. Also, is there the intention of looking at
alternative universes, and still maintaining an idea of the "actual
universe"? It would appear that such issues and other related issues
will be implicitly and explicitly addressed in several of the above
talks. I would very much like to see what directions people want to
take this in and find promising.

DETAILED COMMENTS

1. Forcing, Multiverse and Realism.

Forcing was introduced and viewed as a technical tool by Cohen. He did
not emphasis, and probably did not envision that it had a special
status for f.o.m. However, it is our main tool for the creation of
many models of ZFC. There are assertions in the direction of forcing
being fundamental, many of which have been refuted, some of which have
been proved. Class forcing has been particularly mysterious, posing
many technical issues not present with set forcing, and its status for
f.o.m. is particularly elusive. Vopenka has a celebrated theorem about
a special status of set forcing in a certain context (does not hold
for class forcing). What can we now say about any fundamental status
of forcing?

2. Is there a good argument for mathematical pluralism? `On the
Multiverse Conception of Set.'

By mathematical pluralism, I take this to mean multiple set theoretic
universes. On the other hand, someone working in non set theoretic
foundations - e.g., some brand of categorical foundations - might mean
something quite different here. The most obvious argument is that it
is an obvious way to go in reaction to failures to decide questions
such as CH. Is there a better argument, or is there an interesting way
to elaborate on this argument by failure?

3. Potentialism about set theory. `A Height-Potentialist Multiverse
View of Set Theory.'

There is a well known difficulty in the foundations of set theory to
the effect that - how can we think of ALL ordinals, when just that
very thought gives us a "new ordinal"? This is of course a rejection
of the class/set distinction. If anything is genuine enough to be a
class, why isn't it a set? Arguments like this lead to what I think is
normally meant by "potentialism". The idea is that there is no genuine
idea of "all ordinals". There is a real difficulty in this kind of
view, as there is with the usual view. That is, that there DOES seem
to be a genuine idea of "all ordinals", and in fact when people make
the usual kind of arguments for potentialism, they DO talk about an
idea of "all ordinals" just like I am. How can we make better
arguments for and against potentialism than I am right now? Also, we
can take the potentialism idea into a realm which it historically has
not been taken. Namely, with regard to the power set operation. There
is a very well known potentialist take on the power set of N. This is
called predicativity. The general idea is that at any "time" we have
only countably many subsets of N, and in fact they have been
enumerated. Therefore, we get more if we reflect on them. But it
doesn't make any sense to talk of all countable subsets of N. The
usual trouble is that it DOES seem to make sense to talk about all of
them. And these arguments go back and forth. What is fairly new is the
idea of taking this potentialism seriously not just for the power set
of N but for the power set of anything, and at the same time either
have potentialism for the ordinals or have no potentialism for the
ordinals. In fact, there are four combinations here: yes/no width
potentialistm, and yes/no ordinal potentialism. All kinds of
potentialism that I have seen discussed, have an implicit (maybe
sometimes explicit) notion of what I will now call INTELLECTUAL TIME.
There seems to be some general notion of STAGES that is lurking around
in these discussion. How can we leverage the notion of Intellectual
Time or Stages properly in order to be able to more clearly and
robustly develop various forms of potentialism?

4. `Non-standard analysis as a computational foundation.'

Alternative foundational schemes with some systematic robust idea is
always red meat. An interesting feature of non standardism is both its
drawbacks and its successes. On the drawback side, we know that there
cannot be any preferred or even describable model of nonstandard
analysis in various senses. There is probably more to do along this
negative vein than has been done. The most well known is that there is
no formula which, provably in ZFC, defines a particular example of
various particular kinds of nonstandard models. However, there is a
context in which this is false. One can provably explicitly define
countable nonstandard models of PA, in fact of the true sentences of
PA. We can take a non principal filter on the class of arithmetic
subsets of N, and take the equivalence classes of arithmetic functions
from N into N mod the filter as the points. However, this leads to the
question: how do we divide the arithmetic subsets of N into big and
small? It would be interesting to prove that there is no such way
satisfying certain criteria. I am under some vague impression that the
recursion theorists have touched on this question. A whole other can
of worms concerns the construction of a natural model of PA + not
Con(PA). Even if you allow reasonable recursion theoretic
constructions, I have never seen this done. (Somehow I am reminded of
a probably totally irrelevant question, but is there a connection: it
is open as to whether any two ways of constructing the Rosser sentence
are equivalent). NOW FOR the successes. It appears that all
nonstandard models, in various contexts, share the same properties,
and I don't mean the just first order properties.They all share an
enormous amount of common information when the notion "being standard"
is allowed to be used - allowed to be used!! That is the whole point
of it, anyway. Now this would be clearly true if we are in a context
where all of the admitted nonstandard models are isomorphic to each
other! If my memory serves me right, I think you do get isomorphisms
at least under certain set theoretic assumptions. But when you don't
have isomorphism, you may still have a lot of elementary equivalence.
I think that this talk is mostly concerned with some more down to
earth contexts related to constructively. But still the important
issue of robustness - that what you get is independent of any
particular construction of the non standard objects (e.g.,
infinistesmials) is of paramount importance.

5. A multiverse axiom induction framework.'

Any title with multiverse, axioms, induction, framework, is going to
be flagged here. I am anxious to see what is meant here.

6. Forcing axioms and maximality as the demand for interpretability.'

I have the same kind of positive reaction. See my above concerning the
issue of whether forcing is fundamental or not, in 1.

Harvey Friedman