This post begins a thread in response to a request by Alan Smaill. The
text of his post follows with my response.

However, I intend to make additional posts under this one in another
attempt to present ideas I have had about set theory. During the past
several months I have not had much success with this. But, I can now
discuss some of the matters in terms of partial model theory whereas I
was not able to before.

At any rate, perhaps otherwise off-topic posts involving my ideas about
mathematics can be redirected here so that other posters do not have to
deal with "clutter."


-----------------------
Alan Smaill writes:

> mitch wrote:
>
> > George Greene wrote:
> >
> > <snip>
> >
> > > It would help to recover the original goal.
> >
> > It would. For numerous reasons, you and I started off badly.
> >
> >
> > >
> > > Daryl, you, and I all AGREE that non-COUNTABLE
> > > sets exist.
> >
> > Actually, we do not.
>
> You do not, I take it.

I am uncertain that some concepts in the foundations of mathematics are
meaningful. I cannot agree to the claim with anything resembling an
ontological commitment

>
> > We can agree, however, that the existence of uncountable
> > infinities is a basic belief of mathematicians since Cantor's
reputation was
> > given new life. Moreover, we can agree that any resolution to any
apparent
> > paradoxes or open questions in the foundations of mathematics must
respect
> > these basic beliefs--their general utility being accepted as given.
>
> Why must you (not we) respect that?

In the spring of 1986 I experienced a compelling mathematical
intuition concering the identity predicate which I believed resolved the
continuum hypothesis. The words I used at the time were "equivalence
attachment."

Despite the poor choices I made then, I subsequently did what I was
supposed to do. I formalized my intuitions in a first-order syntax. As
a shorthand, let me refer to the sentences of this first investigation
as [mitch's extensional Sigma] or [meS]. I include the term
'extensional' because

[meS] |= (the usual axiom of extension in ZF)

At least, I believe this consequence relation holds. I have been unable
to find anyone to verify my proofs. If it does hold, then every model
of [meS] is an extensional class. But it is not even clear to me that
such an assertion makes sense.

For most people interested in set theory, ontological commitment has
something to do with the axiom of infinity or large cardinal axioms. My
formal mathematical training barely exceeds an undergraduate course of
study. I was enrolled for three weeks in a course on set theory before
having to drop it for health reasons. A person with such a limited
background is forced to be prudent when faced with the question of what
one might *believe* concerning formal set theory.

So, in all of my research I have always maintained that nothing I
might propose should impact the classical development of mathematics.
Moreover, where I had initially been willing to accept any consequences
for foundational mathematics, I now see the entire problem from a
different perspective.

Prior to my decision to pursue a degree in mathematics, I took a
course on Kant's "Critique of Pure Reason." I recognize that most of my
intuition arises from that influence. I rediscovered Kant after working
out my own ideas while searching for some philosophical justification.
During that part of my research, I concluded that the classical
development of formal systems *as I was aware of them* fell into the
category of analytic inquiry rather than synthetic inquiry.

This had been a key observation. It permits overlap between analytic
philosophy and mathematics without an exclusionary paradigm. But, it
also leaves ontological commitments to conventional paradigms devised in
the twentieth century completely undecided.

All of this can be put much more simply. When one is deciding the
character of one's own beliefs, it is prudent to respect the opinions of
others in areas not yet considered.


> It looks like you have an alternative resolution to the paradoxes.
>
> Why not explain it to us?


I should not go so far as to say that I have resolved paradoxes. But I
can try to explain my sentences again. With the material I have looked
at over the last few months, their presentation may be more interesting
now.

:-)

mitch

In article <3F209917.55D45652@rcnNOSPAM.com>,
mitch <mitchs@rcnNOSPAM.com> wrote:
> In the spring of 1986 I experienced a compelling mathematical
>intuition concering the identity predicate which I believed resolved the
>continuum hypothesis.

One thing I have been wondering is what you mean by "resolving the continuum
hypothesis." Here are some possibilities:

1. First-order logic, the formal axioms of ZFC, and the standard
formalization of the continuum hypothesis into a formal sentence in
the first-order language of set theory are all unproblematic, but
there is a mistake in the Goedel/Cohen relative consistency proof
and CH (or ~CH?) is a formal theorem of ZFC. I doubt that this is
what you mean.

2. First-order logic and the formal axioms of ZFC are unproblematic, as is
the Goedel/Cohen proof, but the standard formalization of the continuum
hypothesis is "wrong" and if CH is properly formulated then it can be
(dis)proved in ZFC. I'm even more doubtful that this is what you mean,
but if it is, then there will be probably lots of debate over what you
have proved is "really" the continuum hypothesis.

3. First-order logic and the standard formalization of CH are unproblematic
but ZFC is problematic, and if ZFC is replaced with a "better" first-order
system of axioms for set theory, then CH is (dis)provable. If your new
axioms are inequivalent to ZFC and sufficiently interesting, then people
will probably like your result, but may not describe it as "resolving" CH,
just as people don't usually think of the fact that V = L implies GCH as
"resolving" CH.

4. First-order logic has to go, and if one adopts the right logic and the
right axioms, then CH can be (dis)proved. If the proof you give can't be
translated into one of the above three possibilities, even in principle,
then there will probably be a lot of debate as to whether your proof is
really a proof.

Which is it?
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

tchow@lsa.umich.edu wrote:

> One thing I have been wondering is what you mean by "resolving the continuum
> hypothesis." Here are some possibilities:

....four possibilities

> Which is it?

I have no idea which it is for Mitch, but none
of your four options captures my understanding
of what it is to "resolve" the continuum hypothesis.

Which is:

5. To resolve the continuum hypothesis is to determine
whether CH is true, and preferably also to come up
with a convincing argument supporting the determination.

(CH is true iff (V, mem) satisfies CH, where:
V is the collection of all pure well-founded sets,
mem is the membership relation over V and
CH is the usual formulation of CH in the first order
language of set theory.

As it happens, this is equivalent to:
CH is true iff (V(w+2), mem) |= CH
which saves us worrying about V not being a set.)

--
Roger Jones
rbj at rbjones dot com

In article <3F214D6F.46795533@signa.ture>,
Roger Bishop Jones <see@signa.ture> wrote:
>5. To resolve the continuum hypothesis is to determine
> whether CH is true, and preferably also to come up
> with a convincing argument supporting the determination.

Presumably, though, you would be able to formalize your "convincing
argument"? Then this would be similar to either 3 or 4, depending on
whether first-order logic and first-order set theory sufficed to express
your argument.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

tchow@lsa.umich.edu wrote:

<snip>

>
> 3. First-order logic and the standard formalization of CH are unproblematic
> but ZFC is problematic, and if ZFC is replaced with a "better" first-order
> system of axioms for set theory, then CH is (dis)provable. If your new
> axioms are inequivalent to ZFC and sufficiently interesting, then people
> will probably like your result, but may not describe it as "resolving" CH,
> just as people don't usually think of the fact that V = L implies GCH as
> "resolving" CH.
>

Thank you very much for this selection Tim. I believe choice number 3 is the
best description.

I also agree that it is unlikely that anyone would describe it as resolving the
continuum hypothesis. Ultimately, provability must be the criterion, and, the
work by Woodin to which you directed me will (hopefully) lead to a hard-won
understanding one way or another.

What I really believe now is that I had some sort of intuition about
constructibility (I had no knowledge of the constructible universe or forcing at
the time). My naive response was to formulate ideas along the lines of
situation theory and mereology within the constraints of set theory (as a
mathematical topic of investigation rather than as an object language).

I should get one or two subposts with a more refined analysis done by the end of
the weekend. And, I know that I need to break them down into starting points
for manageable discussions this time.

Thanks again.

:-)

mitch

tchow@lsa.umich.edu wrote:
> In article <3F214D6F.46795533@signa.ture>,
> Roger Bishop Jones <see@signa.ture> wrote:
>>5. To resolve the continuum hypothesis is to determine
>> whether CH is true, and preferably also to come up
>> with a convincing argument supporting the determination.
>
> Presumably, though, you would be able to formalize your "convincing
> argument"? Then this would be similar to either 3 or 4, depending on
> whether first-order logic and first-order set theory sufficed to express
> your argument.

Let's say it does. It's *still* not equivalent to 3, because you
haven't said anything in 3 about convincing people that your new
axioms are true.

tchow@lsa.umich.edu wrote:
>
> In article <3F214D6F.46795533@signa.ture>,
> Roger Bishop Jones <see@signa.ture> wrote:
> >5. To resolve the continuum hypothesis is to determine
> > whether CH is true, and preferably also to come up
> > with a convincing argument supporting the determination.
>
> Presumably, though, you would be able to formalize your "convincing
> argument"? Then this would be similar to either 3 or 4, depending on
> whether first-order logic and first-order set theory sufficed to express
> your argument.

I didn't stipulate that the argument must be formalizable,
and this is quite deliberate.

Any formalizable resolution of CH effectively reduces the
truth of CH to the truth of some other proposition which
must be at least as uncertain as CH itself.

The resolution of CH therefore depends upon making
informal arguments which are convincing.

In my opinion, it is crucial that any statement of the
problem makes clear that it is ultimately the truth
of CH which is at stake, (and must therefore reach
first base in making clear what it means).

An example of an argument which may not be formalizable
is that CH follows from Woodin's conjecture:

Every set theory that is compatible with the
existence of large cardinals, and makes the
properties of sets with hereditary cardinality
at most aleph-1 invariant under forcing,
implies that CH is false.

I think your presentation of the possibilities
for resolving CH suggests without stating:

(a) that any resolution must be a formal
derivation and that it will leave open
the problem of justifying the formal
system within which the derivation
takes place.

(b) that ultimately this makes the truth
of CH arbitrary and a genuine resolution
unlikely

However it is possible to accept that no formal
resolution is possible without denying that

(a) CH has a definite objective truth value
(b) We may ultimately understand set theory
well enough that there is a reasonable
degree of consensus on what that truth
value is.

The source of our possible knowledge in this
area is our informal understanding of what
"V" is, i.e. of the concept of extensional,
pure, well-founded set.

This concept, like the concept of natural
number, is not susceptible of complete
formalization, but is nevertheless reasonably
clear, has provided the basis for the
development of set theory to date, and may
yet allow us to decide whether CH is true.

--
Roger Jones
rbj at rbjones dot com

Mike Oliver wrote:

> tchow@lsa.umich.edu wrote:
> > In article <3F214D6F.46795533@signa.ture>,
> > Roger Bishop Jones <see@signa.ture> wrote:
> >>5. To resolve the continuum hypothesis is to determine
> >> whether CH is true, and preferably also to come up
> >> with a convincing argument supporting the determination.
> >
> > Presumably, though, you would be able to formalize your "convincing
> > argument"? Then this would be similar to either 3 or 4, depending on
> > whether first-order logic and first-order set theory sufficed to express
> > your argument.
>
> Let's say it does. It's *still* not equivalent to 3, because you
> haven't said anything in 3 about convincing people that your new
> axioms are true.

Astute.

I have always understood the persuasion requirement, if it may be called
that. Of course, the first person you must persuade is yourself.

Subsequently, however, you are faced with belief systems that may seem
impenetrable. The situation is not so terrible if you can associate your idea
with a concept having modest utility. There is leverage in such a
circumstance.

It may be the case, however, that such a turnkey concept is lacking. One may
then be faced with what Barry Smith calls a non-relational accident,

"In the realm of accidents, however, we can
draw another sort of distinction that offers a
parallel to the distinction between collectives
and individual substances. This is the distinction
between relational accidents on the one hand
and non-relational (or one-place) accidents on
the other. Non-relational accidents are
attached, as it were, to a single carrier, as a
thought is attached to a thinker."

In contrast, the persuasion requirement involves a transformation to
relational accidents,

"Accidents are relational if they depend upon a
plurality of carriers and thereby join the latter
together into complex wholes of greater or
lesser duration."

However it occurs, there must be a jump from "one" to "more than one."

Tim Chow once noted that I might expect to provide philosophical justification
for my ideas. This is fundamentally different from proofs invoking axiomatic
premises. If you look into theories of justification, you will find that
there is some analysis resolving the concept of a basic belief to an
infinitist hierarchy of basic beliefs, although it is not characterized as
such.

The persuasion requirement is a significant hurdle because it involves
beliefs.

:-)

mitch