Physical experiment to test continuum hypothesis?
David Ullrich
>A platonist would say that while we don't know whether the continuum
>hypothesis is true or false, and while ZFC can't tell us the answer,
>there is a fact of the matter to be had here. Others would dismiss
>this and say instead that we are free to choose whether the continuum
>hypothesis is true or false, since neither way leads to contradiction.
>
>It occurred to me that there is a kind of "physical interpretation"
>of the statement that ZFC is consistent: namely, it says that a machine
>programmed to enumerate all theorems of ZFC will never spew out a
>contradiction. We can build such a machine and watch it run; of course
>we will die and the machine will fall apart before a contradiction ever
>appears, leaving us none the wiser, but somehow watching the machine
>run makes it easier to believe that there is a fact of the matter
>about whether ZFC is consistent. After all, either the machine will
>eventually spit out a contradiction, or it never will.
>
>My question is whether there is a similar "physical interpretation" of
>the continuum hypothesis. Is there some program that we can set running
>such that the question of whether it ever terminates is equivalent to
>the question of whether the continuum hypothesis is true?
>
>Probably the answer to this question is obvious, but I don't see it
>offhand.
I don't see any reason to believe that the question "is CH true" has
a well-defined answer. If ZF is inconsistent we need to revise everything,
so let's assume for a second that ZF is consistent. Then we know that
ZFC+CH and ZFC+-CH are both consistent. They both have models, and I don't
see any reason to prefer one over the other.
CH is either true or false if you're an actual Platonist, but there
aren't many logicians around who would admit to actual Platonism. Wiener
is a prominent example, a loud example anyway, but if you glance at sci.logic
you see that not everybody always agrees with everything he says.
I don't think that the question of the truth or falsity of CH is
well-defined, and neither do a lot of people. If you revise the question
to "is there a physical test for the consistency of ZFC+CH?" then the answer
is yes, it's exactly the same as for ZF: Program a Turing machine to write
down all the consequences of ZFC+CH one by one and watch to see whether a
contradiction ever pops out.
Of course an equivalent method would be to program the TM to write
down the consequences of ZF. If you think that these things are actually
consistent this might be a better plan because the TM will be easier to
write - on the other hand if you think they're inconsistent then using
ZFC+CH might be better, because the contradiction might happen sooner
given more raw material. My advice would be not to hold your breath
waiting for that contradiction.
--
David Ullrich
Don't you guys find it tedious typing the same thing
after your signature each time you post something?
I know I do, but when in Rome...
Timothy Chow
certain consistency statements, such as Con(ZFC), are independent of ZFC.
Is it true that for any formal system X such that Con(X) is independent
of ZFC, the continuum hypothesis is independent of ZFC + Con(X)?
--
Tim Chow tyc...@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949
David Ullrich
>Let me state my question more precisely. Assume ZFC is consistent. Then
>certain consistency statements, such as Con(ZFC), are independent of ZFC.
>Is it true that for any formal system X such that Con(X) is independent
>of ZFC, the continuum hypothesis is independent of ZFC + Con(X)?
Uh, formal systems are not consistent or inconsistent, theories
within a formal system are consistent or inconsistent. (Not that it's actually
important to get the definitions straight before we start proving things...)
If you change the question to one about "...any theory X in the language
of set theory..." then the question makes sense. I don't know the answer, but I'd
be astonished if the answer was yes.
Moses Klein
Timothy Chow <tc...@math.lsa.umich.edu> wrote:
>Let me state my question more precisely. Assume ZFC is consistent. Then
>certain consistency statements, such as Con(ZFC), are independent of ZFC.
>Is it true that for any formal system X such that Con(X) is independent
>of ZFC, the continuum hypothesis is independent of ZFC + Con(X)?
Yes. Con(X) is a statement of first-order arithmetic stating that there
does not exist a natural number N which is the Godel-number of a proof in
X of a contradiction. The usual methods for proving independdence in ZFC
(e.g. forcing, which can be used to show CH both unprovable and
irrefutable, and Godel's L, which can be used to show CH irrefutable)
produce models with the same natural numbers as the original model, and
hence the same truths of first-order arithmetic. So assumptions about
first-order arithmetic cannot block either Cohen's or Godel's proofs.
Moses Klein (kl...@math.wisc.edu)
Bill Taylor
|>
|> I don't see any reason to believe that the question "is CH true" has
|> a well-defined answer.
I agree 100%. (Well... 98.5%)
|> CH is either true or false if you're an actual Platonist, but there
|> aren't many logicians around who would admit to actual Platonism.
There are more who are, than who would admit to it, though, I'm sure.
|> Wiener is a prominent example, a loud example anyway,
<snigger>
|> but if you glance at sci.logic
|> you see that not everybody always agrees with everything he says.
Folks, we have a winner for the "Understatement of the Year" award!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|> I don't think that the question of the truth or falsity of CH is
|> well-defined, and neither do a lot of people.
Yes, perhaps even a majority, of those who consider the matter.
As I said in another post just now, though realist about PA and uncertain
about Analysis, I'm definitely mostly formalist about ZFC. My guess is this
is quite a common stance, though seldom expressed. Trouble is, if one wants
to give *some* sort of definite meaning to CH and similar statements, then
it often seems to me that the only way is to stick with *Definable Sets*,
which have a clear-cut meaning. But these do not (appear to) satisfy ZFC, only
ZF. Then the question of CH becomes even more vexed; for there are different
ways to word it, (which are equivalent in ZFC but non-equivalent in ZF), and
one is true of definable sets, and one false.
So for those stuck on ZFC, you've either got to be an out-&-out platonist, or
else put up with a permanent problem - either CH has no meaning at all, or
else it's semantically ambiguous.
My choice is to drop AC, which we all know in our hearts is false anyway... ;-)
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
If anyone disagrees with anything I say, I am quite prepared
not only to retract it, but also to deny under oath that I ever said it.
-------------------------------------------------------------------------------
David Ullrich
else put up with a permanent problem - either CH has no meaning at all, or
else it's semantically ambiguous.>
I guess I don't understand what "meaning" means - my view is that it's
neither true nor false, it's true in some models of set theory and false in
others, so what? The statement "xy=yx for all x,y" is true in some models of
first-order group theory (ie in some groups) and false in others, I don't see
how that makes it meaningless.
It's a problem if you believe there's a "real" model of ZF out there
somewhere, but I don't see that as any more sensible than supposing that there's
a real group sitting somewhere in space, and to answer the question of whether
xy is really the same as yx we need to refine our understanding of the real
group somehow.
But I've decided it's a mistake to try to figure out the meaning of the
word "meaning". Whatever it is, CH is not useless just because we don't know
whether it's true or false and disagree over whether that question even means
something - curiously, we don't need to know whether it's true or false for it
to be useful. For example, if you can construct an example with certain properties
by assuming CH then although that doesn't convince a person that such an example
really exists, it does indicate that it's a waste of time to attempt to disprove
the existence of the example (unless you're prepared to begin by rejecting ZF).
Moses Klein
David Ullrich <ull...@math.okstate.edu> wrote:
>interpret sentences ending with the character "?" as questions:
>
> Are there some hidden assumptions here? (For example would all this
>be strictly correct if X were a theory in some uncountable language?
The Godel technique for interpreting Con(X) as a statement of arithmetic
only works if X is recursive, and therefore in a countable language. I
suppose working in set theory we could define Con(X) for a much larger
class of theories X, including some in uncountable languages. X is
consistent iff it has a set model. This would not yield a statement of
arithmetic, so my point above would not apply. I guess that does count as
a hidden assumption. But let's bring this back to Tim Chow's original
question. We want a physical experiment to give a partial decision
procedure. How could you construct a machine to generate theorems in an
uncountable language?
> Do we
>sort of assume that the language is countable as the default, or do we say
>that it doesn't matter because any given proof could be given in some
>countable sub-language or what?)
We can't have a recursive theory in an uncountable language, and if the
theory isn't recursive, no Turing machine can generate its consequences.
>just as a check on whether I'm following this: Could it be that your
>"Con(X) is a statement of first-order arithmetic..." should really be
>"Con(ZFC+Con(X)) & Con(ZFC+NotCon(X)) is a statement of first-order
>arithmetic..."?
No. Here is what Tim Chow speculated: Some statement of the form Con(X)
decides CH. I say, work in a ground model of ZFC+Con(X). We can then
force CH without affecting the truth of Con(X), because Con(X) is a
statement of arithmetic and forcing will not change that. We can likewise
force not-CH without affecting the truth of Con(X). So Con(ZFC+Con(X))
implies both Con(ZFC+Con(X)+CH) and Con(ZFC+Con(X)+not-CH), i.e. Con(X)
does not decide CH (over ZFC). In this argument it is the absoluteness of
Con(X), rather than of the longer consistency formulas, that is relevant.
Incidentally, since Con(X) is a Pi_1 statement of arithmetic (a statement
of the form (for all x)(some primitive recursive predicate), if it is
consistent with PA it is true in the standard model.
Moses Klein (kl...@math.wisc.edu)
David Ullrich
>decides CH. I say, work in a ground model of ZFC+Con(X).
OK. It appeared to me that you were taking his hypothesis to be Con(X),
when in fact he wasn't supposing Con(X), he was supposing the independence of
Con(X). But (duh) if Con(X) is independent of ZFC then yes, ZFC+Con(X) has a
model - I didn't realize that's where you were starting, duh.
Duh^2: OK, Con(X) obviously follows from the independence of Con(X).
Ilias Kastanas
In fact, more than that is true: there are situations where you can
use CH as a hypothesis "for free".
If P is an arithmetical statement (first-order number theory), and
there is a proof of P from ZF + CH (or ZF + AC, or ZF + AC + CH + GCH, or
others that I'll skip), then there is a proof of P from ZF. The latter
proof can be effectively obtained from the former.
So you don't really need CH etc., but there is no loss in assuming it
if you want to; it might facilitate your argument. There is no mathematical
need; it is psychological! You might of course annoy some people and incur
their wrath...
Ilias
Bill Taylor
|>
|> I guess I don't understand what "meaning" means - my view is that it's
|> neither true nor false, it's true in some models of set theory and false in
|> others, so what? The statement "xy=yx for all x,y" is true in some models of
|> first-order group theory (ie in some groups) and false in others, I don't see
|> how that makes it meaningless.
PEEP! Foul ball!
Come on Dave - this won't do. One can't meaningfully compare theories which
are INTENDED to have more than one model, like group theory, ring theory, etc,
with those that are intended to have only one ("real") model; perhaps one 2nd
order model in fact. Things like PA, ZFC, EucG2, and so on.
|> It's a problem if you believe there's a "real" model of ZF out there somewhere
Yes, that's the whole point. The "intended" model. Unfortunately, the intended
model for ZF(C) is a lot more obscure to the intuition than that for PA etc.
This is a permanent problem to ZF-platonists, (except those who insist they
can "see it" without let, hindrance or doubt; like weemba).
|> CH is not useless just because we don't know
|> whether it's true or false and disagree over whether that question even means
|> something - curiously, we don't need to know whether it's true or false for it
|> to be useful. ... if you can construct an example with certain properties by
|> assuming CH then although that doesn't convince a person that such an example
|> really exists, it does indicate that it's a waste of time to attempt to
|> disprove the existence of the example
RIGHT ON! I make this point in "is AC true?" debates - in some ways it makes
it even *more* important to know it's independent. Because then there's *no*
geting round the things proved using it (or its negation), they will be outright
undisprovable, so (as you say) don't waste your time trying to disprove them.
Still, it does leave these things (like AC) up in the air, as a matter of faith.
Cheers.
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Faith is believing what you know ain't so. -Huckleberry Finn
-------------------------------------------------------------------------------
David Ullrich
>In article <460s7t$d...@tiber.uoknor.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>>by assuming CH then although that doesn't convince a person that such an example
>>really exists, it does indicate that it's a waste of time to attempt to disprove
>
>
> In fact, more than that is true: there are situations where you can
> use CH as a hypothesis "for free".
>
> If P is an arithmetical statement (first-order number theory), and
> there is a proof of P from ZF + CH (or ZF + AC, or ZF + AC + CH + GCH, or
> others that I'll skip), then there is a proof of P from ZF. The latter
> proof can be effectively obtained from the former.
>
> So you don't really need CH etc., but there is no loss in assuming it
> if you want to; it might facilitate your argument. There is no mathematical
> need; it is psychological! You might of course annoy some people and incur
> their wrath...
>
(Are there any "real" theorems of first-order PA the proof of which
is substantially easier given CH?)
David Ullrich
>David Ullrich <ull...@math.okstate.edu> writes:
>|>
>|> I guess I don't understand what "meaning" means - my view is that it's
>|> neither true nor false, it's true in some models of set theory and false in
>|> others, so what? The statement "xy=yx for all x,y" is true in some models of
>|> first-order group theory (ie in some groups) and false in others, I don't see
>|> how that makes it meaningless.
>
>PEEP! Foul ball!
>
>Come on Dave - this won't do. One can't meaningfully compare theories which
>are INTENDED to have more than one model, like group theory, ring theory, etc,
>with those that are intended to have only one ("real") model; perhaps one 2nd
>order model in fact. Things like PA, ZFC, EucG2, and so on.
>
(Or maybe the fact that ZF is intended to have only one model doesn't
quite make it so. I'm never quite certain what the "real" model of ZF is (unless
we're talking V=L, which I don't think was the intent). Years ago it would have
been clear to me what the real model was, but some years before that it was
clear to everybody what the real model of naive (ie inconsistent) set theory
was. Not that I'm going to, but if gave you two models of set theory could you
tell me what the real one was? I'm not asking for an effective procedure or
anything, what's the theoretical distinction?)
john baez
>David Ullrich <ull...@math.okstate.edu> writes:
>|> I guess I don't understand what "meaning" means - my view is that it's
>|> neither true nor false, it's true in some models of set theory and false in
>|> others, so what? The statement "xy=yx for all x,y" is true in some
>|>models of
>|> first-order group theory (ie in some groups) and false in others, I
>|>don't see how that makes it meaningless.
>PEEP! Foul ball!
>Come on Dave - this won't do. One can't meaningfully compare theories which
>are INTENDED to have more than one model, like group theory, ring theory, etc,
>with those that are intended to have only one ("real") model; perhaps one 2nd
>order model in fact.
But maybe we should change our attitude and stop wanting there to be
one "real" model of things like PA or ZFC. It might just be a kind
misguided "fundamentalism" to think that there is a "real" model. What
good, exactly, does it do us to think that there is one model that's
"better" than the rest, in a theory that has lots of models?
Timothy Chow
>But maybe we should change our attitude and stop wanting there to be
>one "real" model of things like PA or ZFC. It might just be a kind
>misguided "fundamentalism" to think that there is a "real" model. What
>good, exactly, does it do us to think that there is one model that's
>"better" than the rest, in a theory that has lots of models?
Indeed; this is why it is misleading to think of platonism as the quest for
the one "real" model. As far as actually doing mathematics goes, platonists
and formalists are indistinguishable. In particular, platonists are perfectly
happy to deal with different models without passing judgment on which one is
the "right" or "better" one. Goedel only claimed that he thought that the
continuum hypothesis was either true or false (i.e., not "neither"); he
didn't claim to know which was the case, and he didn't play favorites when
it came to studying ZFC + CH and ZFC + ~CH.
What platonists claim is that mathematical assertions, and in particular
mathematical assertions about sets, are cognitive, i.e., matters of fact
that may be either true or false but not "neither" or "both" (in contrast
with "Ouch!" or "Which is the way to San Jose?" which are non-cognitive).
Now, it follows from this that certain models of formalized set theory
will be more "realistic" in the sense that true statements about sets will
also be true in those models. But platonists don't say this because it
"does them any good" or because they are seeking the Holy Grail of the
"real" model. They say it because it follows from the claim that assertions
about sets are cognitive.
Consider PA. By calling one of the models of PA the "standard" one, we are
not saying that other models of PA are inferior. The reason we single one
out as being "standard" is that we have a clear idea in our minds about what
the integers are, and we believe that statements such as "every even integer
greater than two is the sum of two primes" are either true or false. It
follows that there is some model of PA whose truths coincide with the actual
truths about the integers. Does it "do us any good" to regard other models
of PA as inferior? Of course not, but we're not doing that. The "misguided
fundamentalism" is a straw man.
--
Tim Chow tc...@umich.edu
David Ullrich
Um, a lot of people would tend to feel that PA and ZF are very different
in this regard. I have an idea what the standard model of PA is: It contains
1, and 2, etc. Of course I can't define what the "etc" means precisely, except
via an axiomatization like PA, but I do feel I know exactly what the model
is, whether I can define it or not. Maybe it's just my inferior brain, but I
don;t have anything like such a clear conception of what a model of ZF could
be.
Which is not to say I don't have a well-developed intuitive notion of the
meaning if the word "set". Alas my intuitive notion of the word "set" is known
to be inconsistent nonsense.
David Ullrich
[snip]
>BT: Yes, that's the whole point. The "intended" model. Unfortunately, the
>BT: intended model for ZF(C) is a lot more obscure to the intuition than
>BT: that for PA etc. This is a permanent problem to ZF-platonists, (except
>BT: those who insist they can "see it" without let, hindrance or doubt;
>BT: like weemba).
>
>DU: Not that I'm going to, but if gave you two models of set theory could you
>DU: tell me what the real one was? I'm not asking for an effective procedure or
>DU: anything, what's the theoretical distinction?)
>
>I think the discussion above represents a misunderstanding of what platonism
>is. The issue is not about models.
Who are you to say what THE issue is? If I ask about the distinction
between the real model of ZF and the fake ones that's an issue, and THAT issue
is about models.
>Again, we need to be careful to avoid a
>Skolem's paradox error: let's use "the continuum hypothesis" to refer to the
>assertion that every uncountable subset of the reals has the same cardinality
>as the reals, and let's use "CH" to denote the formalization of this statement
>in ZFC. A platonist would affirm that CH is true is some models of ZFC and
>false in others. But it does not follow from this that the continuum
>hypothesis is neither true nor false. The platonist would deny this, and
>assert that *either* every uncountable subset of the reals has the same
>cardinality as the reals, *or* not every uncountable subset of the reals
>has the same cardinality as the reals. One or the other is the case.
>It is also somewhat misleading to characterize the issue as the question
>of whether or not there is a "real" model of ZF out there, or what the
>"intended" model is. It is clearer to say, "it's a problem if you believe
>there *really are* sets out there" and "sets are a lot more obscure to the
>intuition than integers are."
>
>To answer Ullrich's "will the real model please stand up" question, let's
>replace CH with Con(ZFC) for the sake of clarity.
It's happened a few times that I've accused you of taking my
questions/comments, revising them, and answering the revised version. Tell
me again that this is not what you're doing.
>If you were to give me
>two models, one of ZFC + Con(ZFC) and one of ZFC + ~Con(ZFC), then I would
>say that the "real" one was ZFC + Con(ZFC).
If I might be so bold as to suggest that there's more than one
question in the universe: Concerning the question I was asking I acually
don't see the difference between CH or Con(ZFC) or whatever. But when you compare
a model of ZFC+~Con(ZFC) with a model of ZFC+Cin(ZFC) you're missing the
point (where "the point" is not meant to suggest that it's the only point,
rather "the point to the question" - I do know what the point to the question was,
it was my question)
If we replace CH by Con(ZFC), ("for clarity" ??????) then A question
is: Suppose I give you two models of ZFC+Con(ZFC). Tell me what the real one
is. (Again, lest we misunderstand, I'm not asking for anything effective, just
the definition that serves to distinguish the two, regardless of whether we can
see which one satisfies the definition.)
>The reason is that I believe
>that ZFC *really is* consistent. Now, I can't give you an effective
>procedure for showing this; fortunately you didn't ask for one. :-) One
>way to see the "theoretical distinction" is via "physical experiment": set
>a machine programmed to enumerate all theorems of ZFC running. If the
>machine eventually spits out a contradiction, then ZFC *really is*
>inconsistent; if it never will, ZFC *really is* consistent.
>
>This example was easy, since most people share the intuition that there is
>a *fact of the matter* about whether the machine will or will not eventually
>spit out a contradiction. (Beware, though, that a strict finitist would not
>share this intuition!) The outcome of this experiment, if we could "see" to
>the end of time, is what would determine which of the two models was the
>"real" one.
>
>The continuum hypothesis is trickier; that's what motivated me to start
>this thread. The simplistic approach for "ZFC is consistent" won't work,
>so it's not as clear (to most of us anyway) what the continuum hypothesis
>means. But it's not inconceivable that some kind of "infinite physical
>experiment" would determine the truth conditions of the continuum
>hypothesis, if uncountable infinities really exist out there in outer
>space or down there in the infinitesimal fabric of spacetime.
>
>One final remark: in the last three paragraphs I took the timid approach
>of using the physical universe (which most people accept is "real" in the
>sense that assertions about it have a definite truth value) to give meaning
>to the claim that certain mathematical statements have a definite truth
>value.
If you want to talk about the "real" reals in mathematics that's
one thing, although I don't believe you're going to be able to tell me how
to distinguish them from fake reals. But the idea that there's something
in the actual physical universe that corresponds closely enough to the reals
to allow one to say whether CH holds in the physical universe is sheer nonsense.
At the very best it's the sort of thing which is intrinsically unknowable and
hence a silly topic for discussion.
(At first "Physical experiment to test CH?" looked like nonsense -
then I decided that the "physical" part was just some physical Turing machine,
fine. But it looks like you actually think it makes sense to ask whether CH
is actually true in the real physical universe? This is incredible - does
anybody else you know think this?)
>It would actually be more in the spirit of Plato to argue that
>mathematical "reality" is actually *more* real than material "reality."
>I avoided this because I thought it would be less convincing, but bear in
>mind that other platonists might have a different account of why it is
>that certain mathematical statements have a definite truth value (i.e.,
>they would give a different "theoretical distinction").
My gosh, when we talk about "Platonism" I would have thought we
were talking about a modern school of thought that sort of had its origins
in some of Plato's work, perhaps. Why in the world would it make any difference
whether something was actually in the spirit of Plato? We're regarding him as
an authority on anything at present? This is incredible.
Regardless, talking about a "real" mathematical universe strikes me
as at least marginally less silly than looking for actual physical objects
corresponding to the reals, etc.
David Ullrich
>In article <46cd9d$d...@cantua.canterbury.ac.nz> w...@math.canterbury.ac.nz (Bill Taylor) writes:
>>David Ullrich <ull...@math.okstate.edu> writes:
>
>>|> I guess I don't understand what "meaning" means - my view is that it's
>>|> neither true nor false, it's true in some models of set theory and false in
>>|> others, so what? The statement "xy=yx for all x,y" is true in some
>>|>models of
>>|> first-order group theory (ie in some groups) and false in others, I
>>|>don't see how that makes it meaningless.
>
>>PEEP! Foul ball!
>
>>Come on Dave - this won't do. One can't meaningfully compare theories which
>>are INTENDED to have more than one model, like group theory, ring theory, etc,
>>with those that are intended to have only one ("real") model; perhaps one 2nd
>>order model in fact.
>
>one "real" model of things like PA or ZFC. It might just be a kind
>misguided "fundamentalism" to think that there is a "real" model. What
>good, exactly, does it do us to think that there is one model that's
>"better" than the rest, in a theory that has lots of models?
Right. Except the "change" part - a lot of people already take this
attitude. (I hope Hilbert doesn't have any internet access, the poor guy
is spinning in his grave if he does...)
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
> Who are you to say what THE issue is? If I ask about the distinction
>between the real model of ZF and the fake ones that's an issue, and THAT issue
>is about models.
By saying that the issue is not about models, I do not mean that the issue
does not involve models at all, but that the *crucial point* in this issue
is not a point about models---it is a point about whether mathematical
statements have a determinate truth value. That's all.
> It's happened a few times that I've accused you of taking my
>questions/comments, revising them, and answering the revised version. Tell
>me again that this is not what you're doing.
I am doing it, but because I think that the answer to the revised question
sheds light on the original question by eliminating irrelevant details.
> If we replace CH by Con(ZFC), ("for clarity" ??????) then A question
>is: Suppose I give you two models of ZFC+Con(ZFC). Tell me what the real one
>is. (Again, lest we misunderstand, I'm not asking for anything effective, just
>the definition that serves to distinguish the two, regardless of whether we can
>see which one satisfies the definition.)
O.K., suppose you give me two models of ZFC + Con(ZFC). You want to know
which is the real one. Well, if the two models are identical, then there
is nothing to say; therefore, assume that the two models are distinct.
Then there is a statement whose formalization is true in one model (call
it M) and false in the other (call it M'). The statement might be "there
exists an uncountable subset of the reals that cannot be surjected onto
the reals," or it might be something else, but let's take this statement
for the sake of concreteness. If there really does exist an uncountable
subset of the reals that cannot be surjected onto the reals, then M is
the real model; if there does not exist such an uncountable subset, then
M' is the real model.
Probably your objection will be, what do you mean, "there really does
exist an uncountable subset of the reals"? You would probably question
the meaningfulness of such an assertion. For this reason, I chose to
consider the situation where you give me two models of ZFC (instead of
two models of ZFC + Con(ZFC)), with the statement whose formalization
is true in one model and false in the other being "ZFC is consistent"
(as opposed to "there exists an uncountable subset of the reals that
cannot be surjected onto the reals"). The reason is that "ZFC *really
is* consistent" clearly has a determinate truth value if any mathematical
statement does, so one cannot reasonably make the objection that "'ZFC
really is consistent' is not meaningful, or at least is neither true nor
false; it's true in some models and false in others---so what?"
>But the idea that there's something
>in the actual physical universe that corresponds closely enough to the reals
>to allow one to say whether CH holds in the physical universe is sheer nonsense
>At the very best it's the sort of thing which is intrinsically unknowable and
>hence a silly topic for discussion.
What's your justification for this claim?
>But it looks like you actually think it makes sense to ask whether CH
>is actually true in the real physical universe? This is incredible - does
>anybody else you know think this?)
It may not make sense, but it is not a priori obvious that it *cannot* make
sense. Physics, after all, proceeds by constructing mathematical theories
that model reality. If someone constructs a physical theory that uses CH
and someone else constructs a rival theory that uses ~CH and the two
theories predict different outcomes, doing the experiment would provide
evidence for one theory over the other. This is a far cry from giving
evidence that CH is "actually true" or that something out there "actually
corresponds" to the reals, but it would at least give some *evidence* for
either CH or ~CH, and thus CH would not be *totally* unknowable. Sure,
this is really far-fetched, but I do not see any argument that rules out
*a priori* the possibility that there exists something out there in the
physical universe corresponding to the reals. Certainly, it is unlikely;
but what is your argument that it is impossible, or unknowable?
>My gosh, when we talk about "Platonism" I would have thought we were
>talking about a modern school of thought that sort of had its origins
>in some of Plato's work, perhaps. Why in the world would it make any
>difference whether something was actually in the spirit of Plato? We're
>regarding him as an authority on anything at present? This is incredible.
Who said anything about authority? I was just trying to give a warning that
others might give a different account from the one I was giving. Since you
complain so bitterly about my misreading you, the least you could do is to
extend me the courtesy that you demand of me. I said that such-and-such was
in the spirit of Plato; did I say that because of this, it was true, or that
because of this, we ought to bow to Plato's authority? No. I simply said
that it was in the spirit of Plato, and that therefore others might give an
account different from mine. I see no grounds for your accusation. Let me
point out that in our discussions; *you* have always been the one to start
talking about appeals to authority.
Matthew P Wiener
>In article <46m0f3$r...@news.cis.okstate.edu>,
>David Ullrich <ull...@math.okstate.edu> wrote:
>>But it looks like you actually think it makes sense to ask whether
>>CH is actually true in the real physical universe? This is
>>incredible - does anybody else you know think this?)
>It may not make sense, but it is not a priori obvious that it
>*cannot* make sense.
And indeed, one can construct a local deterministic interpretation
of QM *assuming* CH (or even just Martin's Axiom). This does not
contradict Bell's inequality, since the interpretation does some
clever tricks with non-measurable sets. It's "physics" enough to
have been published in PRD.
See Gudder's book on quantum probability, and the references therein.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
Ilias Kastanas
Bill Taylor <w...@math.canterbury.ac.nz> wrote:
>ika...@alumni.caltech.edu (Ilias Kastanas) writes:
>
>|> You are being too pessimistic here! A ground-level notion of "defi-
>|> nable sets" is L, Goedel's constructible sets.
>
>Aaaarrrggghh! Not a bit! Godel's constructible sets, (how I hate that
>misleading name, though I can't think of a better one off-hand :)), and
>definable sets, are very far apart. Neither includes the other, in fact.
>People are fooled by the name ("constructible sets") into thinking they
>are in some way philosophically canonical, whereas they are merely a
>technical convenience. V=L gets more currency than it deserves this way;
>closer inspection shows it to be just a souped-up form of choice, in disguise.
"Midterm (2 problems):
1. Define the universe.
2. Give two examples."
In the spirit of the above (not really _that_ funny... after all,
V=L is _one_ example, huh?) I would ask you: please define "definable".
No, I'm not trying to spring the 'least non-definable x' on you... I
am trying to find out what notion of "definable" you have in mind, in parti-
cular which definable object is not in L. (I can safely guess objects in L
that you consider non-definable...)
The point is, what hypotheses will you accept to defeat ( definably,
too!) poor V=L... which is, after all, consistent. You are enjoying the
illicit pleasures of measures on Power(R)... would you actually go for
a measurable cardinal?
Goedel's original construction of L might seem ad hoc and technical,
iterating those specific operations. But it is equivalent to the later
approach, each layer having the sets that are first-order definable (with
parameters) below. That does seem canonical! In fact, it is absolute
(unlike "power(omega)" or "greater cardinality than"); it is "the same", no
matter in what model of ZF it is carried out, yielding a sparsest possible
model (very sketchy and rough)
Crass imposition of a global wellordering? Sour gr- uh, OK, but the
point is, one does end up with a model of Set Theory! No other sets are
needed.
What can be described in first-order logic appears to be "definable",
not exotic, and would be hard to avoid. Induction on the ordinals is
routine. What are your objections to L?
[ . . . ]
>|> >My choice is to drop AC, which we all know in our hearts is false anyway... ;-)
>|>
>|> OK, my heart is ready to embrace a family of nonempty sets whose
>|> Cartesian product is empty,
>
>A convert-to-be! Just kneel with me and pray that the Lord will open your eyes!
>
>|> if such can be found. Any suggestions??
>
>Plenty. But I'll let you do the dirty work transferring the statement to
>"Cartesian product of family of nonempty sets" form! Use any one of these...
>
>A non-principle ultrafilter on N.
> (Or as it said in the intelligencer cartoon - "an unprincipled infiltrator")
>A non-measurable subset of R.
Hmm, I get it... "immeasurably real agony..."
>A well-ordering of R.
"... fully regimented police state..."
All right, enough of this. The Greeks suffered the irrationals, and
accepted them. Now you take your medicine too.
>All these are "obvious" counterexamples to AC, and should be easy to translate
>into your form. (!) ;-) Most folk accept your Cartesian-product statement as
>obviously true, but only because they always think of a countable family; and
>countable choice *does* appear to be true.
>
>But I always await a counterexample!
Side comment: Have you ever been confronted with some of the plethora
of pseudo-arguments lurking here? "R is so rich and complex that it cannot
be wellordered... hence it must surely be able to contain irregular, non-
measurable sets..."
Of course, a wellordering of R (or, for that matter, a non-principal
ultrafilter on omega) is exactly what yields, easily, a non-measurable set.
Anyway: the above examples, and even the Banach-Tarski miracle, have
never been 'arguments against AC' for me.
Maybe I consider them less outrageous than you do; but that's not the
reason. I just find AC supported by intuition, by the "nature" of sets.
Nonemptiness of Cartesian products is as natural as subsets, unions and so
on; if the elements are there, the tuple is there too. The contrary seems
unmotivatedly limiting and weird. Countable family or not - no difference;
and if ZF does not capture AC-intuition, so what?
(What a heretic, eh?!)
I don't downplay or disparage effectiveness and definability. We can
study whether definable objects, choice functions, etc. exist or not; should
we resent the presence of 'arbitrary' ones? The notion of function from R
to R seems indicated, even if one only uses continuous, or measurable ones.
Hardly anybody disputes that "there is no definable wellordering of R"
... whatever "definable" may mean. I would have thought this covers, in
substance, your position. But it doesn't; you actually feel it is not
enough, and you have to drop AC anyway. What are your reasons?
Footnote: Funny how the Cartesian
Product form of AC (Russell's, not mine..)
is seen as the "most compelling" one, more
than Choice Function.. even though they say
the same thing!
Ilias
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
> Huh? This is no definition at all: You say that there's some statement
>S which is true in M and false in M', fine. Now you say that if S is true in
>the real reals then M is the real one, while if it's false in the real reals
>then M' must be the real one. What? How do I know whether it's true ot false
>in the real reals when I don;t know what they are yet? What you say is
>certainly true, but as a _definition_ of the phrase "the real reals" it's
>circular, no help at all.
I'm not using this as a definition of "the real reals." You asked for the
distinction between the "real" model of ZFC and a "fake" model of ZFC.
I'm responding that the distinction between the "real" and "fake" models
is grounded in the concept of the "real reals." This is not *circular*,
because the "real model of ZFC" is not the same object as "the real reals"
(models are not numbers, any more than circles are prime ideals). It does
indeed *beg the question*, what do you mean by "the real reals"? (For this,
see below.)
That's why I (rather informally) said that "the issue is not about models."
By staring only at the two models of ZFC, one is never going to be able to
say which one is "real" and which is "fake." One must realize that the
distinction between "real" and "fake" models is grounded in the notion
of the "real reals." I claim that it is easier to make sense of the
notion of the "real reals" than it is to make sense *directly* of the
distinction between "real" and "fake" models.
> Here you explain that if CERTAIN statements are true in M and false
>in M' then you can tell me which is the real one. But what if none of the
>statements that you happen to somehow know the answer to serves to distinguish
>M from M'?
Before going on, let me make sure you agree to the following statement
(because it sounds to me that you do): "ZFC is consistent" is a statement
that has a determinate truth value, i.e., it is *not* "neither true nor
false," in spite of the fact that there are some models of ZFC in which
Con(ZFC) is true and there are others in which Con(ZFC) is false.
If you agree, then let me point out that what we have here is an example
of a statement (namely, "ZFC is consistent") which has a determinate truth
value, *even though* we don't know whether it's true or false, and *even
though* we can prove that it is unprovable from ZFC. Thus (here is the
crucial point) whether or not a statement has a determinate truth value
is *not* dependent on whether or not you have axiomatized the objects
that the statement talks about precisely enough to prove the statement
one way or the other.
Really, that's the crucial point. What makes us so sure that "ZFC is
consistent" is "really" true or false? It's because we believe that
there "really is" such a thing as ZFC, and that we know *what* it is
clearly enough that a statement such as "ZFC is consistent" must, in
our mind, be either true or not true. Either there's an integer "out
there" that codes a proof of a contradiction from ZFC, or there isn't.
Notice, in particular, that our conviction here is *not* grounded in
axiomatization; the way we have axiomatized mathematics still leaves
the question of the consistency of ZFC an "indeterminate" question.
The arguments you have been using in this discussion could be turned
around into an argument that "ZFC is consistent" is "neither true nor
false." I could argue that there are models where Con(ZFC) is true
and others where it is false; if you're going to single out one of
them as being "real" you have to tell me what ZFC "really" is and
what consistency "really means," in such a way that I can distinguish
the "real ZFC" from the "fake" ZFC's which are "fakely inconsistent"
---whose contradictions are encoded by nonstandard integers, for
example. But I don't think that any of this talk makes it any less
clear to me that I know what I'm talking about when I say that ZFC
is *really* consistent or that "ZFC is consistent" is *really* true.
Notice that the question of whether there's a "real" ZFC boils down
to a matter of how sharp our intuition about mathematical objects is
(and *not* to a matter of how sharp our axiomatizations are).
Notice also that I can say that "ZFC is consistent" has a determinate
truth value without knowing which is the case, or having a method for
deciding which is the case. My ignorance would prevent me from saying
*which model* of ZFC was the "real" one, but it would not prevent me
from claiming that one of them *was* "real."
Passing to the general case, someone who claims that there is some
"real" model of ZFC is just claiming that they have a sharp enough
intuition about sets that assertions such as the continuum hypothesis
have a determinate truth value. Such a person can freely admit that
their axioms about sets can't deliver a verdict on the question, but
as we saw in the case of "ZFC is consistent," this doesn't imply that
the continuum hypothesis has no determinate truth value, any more than
the fact that our axiomatizations of concepts such as "a recursive set
of axioms" or "consistency" are insufficiently sharp to deliver a
verdict on the question of the consistency of ZFC implies that "ZFC
is consistent" has no determinate truth value.
Once one has such an intuition, the theoretical distinction between
the "real" and "fake" models of ZFC follows right away: the "real"
model is the one whose truths correspond to the actual facts about
sets (whatever those facts happen to be; we may not have any way of
finding them out).
I think part of the confusion stems from the idea that we don't know
what we're talking about until we axiomatize it, and then somehow the
only statements that make sense are those that the axioms can say
something about. But in fact, our mathematical intuition is what
"makes sense" of mathematical statements; axiomatization comes later.
> Come on. Of course I'm not going to attempt an a priori mathematical
>proof of anything involving the physical universe. But really: You have this
>physical entity which you think is exactly isomorphic to R. How could you
>possibly have any evidence whatever that it's actually isomorphic to R and not,
>say, Q? Or some non-archimedean field? Or anything else that looks just like
>R at any non-infinitesmal scale?
Well, I gave my argument already. One cannot perform a "direct" physical
experiment to tell the difference, but physics isn't just about "direct"
physical experiments. It's about formulating theories and testing
predictions of the theories. I say, if the universe really were more
like Q^3 than R^3, why is it that nobody has ever come up with a theory
that models the universe as Q^3? To someone who believes that the universe
is really more like Q^3, I would say, isn't it odd that even though the
universe is really Q^3, we have trouble pretending that it is, where as we
are wildly successful when we pretend it's R^3? Doesn't prove anything,
of course, since we're just pretending, but doesn't it suggest that the
universe, whatever it is, is really more like R^3 than Q^3?
I don't really care to argue this point very far, because I don't happen
to believe that CH is really embedded out there in the universe waiting
for our theories and experiments to find it. I just don't think that it
can be dismissed as "sheer nonsense."
Matthew P Wiener
>Indeed; this is why it is misleading to think of platonism as the
>quest for the one "real" model. As far as actually doing mathematics
>goes, platonists and formalists are indistinguishable. In
>particular, platonists are perfectly happy to deal with different
>models without passing judgment on which one is the "right" or
>"better" one. Goedel only claimed that he thought that the continuum
>hypothesis was either true or false (i.e., not "neither"); he didn't
>claim to know which was the case, and he didn't play favorites when
>it came to studying ZFC + CH and ZFC + ~CH.
Similarly, formalists have been known to have or favor intuitions regarding
the correct resolution of CH. Perhaps the most famous is Cohen himself in
his little book, where he spells out the view that maybe some day we'll
come to the conclusion that CH is "obviously" false--way way way false--
and that power set formation in general is a humungous undertaking.
Matthew P Wiener
>I'm never quite certain what the "real" model of ZF is (unless we're
>talking V=L, which I don't think was the intent).
There is a wonderful historical joke regarding Goedel and V=L.
At the time, foundational intuitions were still fuzzy. People were
going around saying, "we all believe, this, right? Right?!" What
worked or did not work would get in as seemed natural. And in the
mid-30s, Goedel announced GCH was consistent.
Goedel could have changed mathematical history, the joke goes, had he
anounced GCH was _proven_. "And here, boys and girls, is our spanking
new foundational axiom that does the trick."
Of course, V=L doesn't answer all questions: are there inaccessible,
Mahlo, weakly compact, or ineffable cardinals?
Mark A Biggar
>Well, I gave my argument already. One cannot perform a "direct" physical
>experiment to tell the difference, but physics isn't just about "direct"
>physical experiments. It's about formulating theories and testing
>predictions of the theories. I say, if the universe really were more
>like Q^3 than R^3, why is it that nobody has ever come up with a theory
>that models the universe as Q^3? To someone who believes that the universe
>is really more like Q^3, I would say, isn't it odd that even though the
>universe is really Q^3, we have trouble pretending that it is, where as we
>are wildly successful when we pretend it's R^3? Doesn't prove anything,
>of course, since we're just pretending, but doesn't it suggest that the
>universe, whatever it is, is really more like R^3 than Q^3?
Well, people are trying this. Look at some of the work dome in trying to
emulate Quantum Theory via Cellular Automata. These theories assume that
space and time are really discrete and try to simulate the universe as
a very complex form of "life" game. See comp.theory.cell-automata.
--
Mark Biggar
m...@wdl.loral.com
john baez
>In <470r6u$5...@netnews.upenn.edu> wee...@sagi.wistar.upenn.edu (Matthew P
>Wiener) writes:
>>Similarly, formalists have been known to have or favor intuitions regarding
>>the correct resolution of CH. Perhaps the most famous is Cohen himself in
>>his little book, where he spells out the view that maybe some day we'll
>>come to the conclusion that CH is "obviously" false--way way way false--
>>and that power set formation in general is a humungous undertaking.
>I've always thought the issue boiled down to how one feels about
>Occam's razor. Is the "real" universe the largest possible one that
>is logically consistent, or is it the smallest possible one that
>includes everything we know for sure must exist?
Personally I think that both these universes are "real" to more or less the
same extent, and that they're both interesting topics of study. I guess
that means I'm more of a maximalist than even the maximalist who wants
the largest possible universe... I want *all* possible universes!
David Ullrich
>In article <46ml1t$h...@controversy.admin.lsa.umich.edu>, tchow@math (Timothy Chow) writes:
>>In article <46m0f3$r...@news.cis.okstate.edu>,
>>David Ullrich <ull...@math.okstate.edu> wrote:
>
>>>But it looks like you actually think it makes sense to ask whether
>>>CH is actually true in the real physical universe? This is
>>>incredible - does anybody else you know think this?)
>
>>It may not make sense, but it is not a priori obvious that it
>>*cannot* make sense.
>
>of QM *assuming* CH (or even just Martin's Axiom). This does not
>contradict Bell's inequality, since the interpretation does some
>clever tricks with non-measurable sets. It's "physics" enough to
>have been published in PRD.
>
>See Gudder's book on quantum probability, and the references therein.
I find this almost unbelievable, which of course proves nothing.
Is it possible to give a hint how physics could possibly have anything
to say about CH or conversely?
Jim Walters
There is a delicious tension between Occam's razor and the precept that whatever
isn't forbidden is mandatory. And the mathematics we do is precariously poised
between the two. That we don't take equal interest in all Post formal systems
indicates there is an esthetic at work. Something closely related to Wigner's
unreasonable effectiveness of mathematics.
It could be that the intuitions of mathematicians, which subsequently are useful
to physics and computer scientists, are motivated by some simple predispositions.
For example: there ain't no free lunch. In physics the sentiment is often expressed
as a conversation law. Mathematicians are reminded of the principle every time
they find "short" proofs of difficult problems which assume a *lot* of underlying
machinery. (There was a recent post in sci.math.research concerning topology
that was of this ilk).
Given the relationship between the KMS condition and Tomita-Takesaki modular
theory, you can't help but wonder: why? How did this group of mathematicians
and physics separated by time and interests stumble across a Post formal system
so closely interrelated? What is going on?
I'm working on a review of Jean-Pierre Changeux and Alain Connes' frustrating
*Conversations on Mind, Matter and Mathematics*. Coming to a USENET site near you
soon. :-)
Jim Walters
try...@halcyon.com
Matthew P Wiener
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) wrote:
>>And indeed, one can construct a local deterministic interpretation
>>of QM *assuming* CH (or even just Martin's Axiom). This does not
>>contradict Bell's inequality, since the interpretation does some
>>clever tricks with non-measurable sets. It's "physics" enough to
>>have been published in PRD.
>>See Gudder's book on quantum probability, and the references therein.
>I find this almost unbelievable, which of course proves nothing. Is
>it possible to give a hint how physics could possibly have anything
>to say about CH or conversely?
I once took a graduate physics seminar in C*-algebraic QFT. The second
day, before class, the professor asked me what my background was. When
I told him set theory, he acted irritated and smirkingly asked me to
show him a non-measurable set. I said, sure, but would he first show
me an electron? He laughed.
What it boils down to is one simple question, "what is reality"? We
don't know, but we do have our models. And if you insist on using the
continuum as a substrate, you're of course asking to get bit at some
point. In this case, it happened when Pirani(?) found a devilishly
subtle loophole in Bell's theorem (the one that asserts that locality
and determinism don't mix in QM). One works with non-measurable pieces
that in three dimensions fit together in a measurable way, and thus it
looks just like ordinary QM.
This of course already requires AC in a blatant way. The construction
is subtle enough that apparently more than ZFC is needed. CH (or MA)
is a standard assumption in certain measure theory problems: as one
inducts up to the continuum, one needs subcontinuum sets to have measure
zero along the way.
Bill Taylor
|> Similarly, formalists have been known to have or favor intuitions regarding
|> the correct resolution of CH. Perhaps the most famous is Cohen himself in
|> his little book, where he spells out the view that maybe some day we'll
|> come to the conclusion that CH is "obviously" false--way way way false--
|> and that power set formation in general is a humungous undertaking.
Yes, I remember being struck by that comment in his CH book. Very cute,
and very neat idea.
Is Cohen really a strict formalist? I didn't know that.
Actually, now that I recall, I think his comment was...
"I suspect a view that may eventuially come to be commonly held is..." etc
So: he was speaking of what he thought those (silly silly) platonists would
eventually come to believe; not necesarily that he held the view himself.
As he couldn't, of course, if he were a formalist.
Howbout that?
Bill.
Tal Kubo
Matthew P Wiener <wee...@sagi.wistar.upenn.edu> wrote:
>subtle loophole in Bell's theorem [...] non-measurable pieces that
>in three dimensions fit together in a measurable way, and thus it
>looks just like ordinary QM.
Fascinating. However, both the question and the disbelief (not to mention
the subject of this thread) had to do with the possibility of CH impinging
on, say, a physical measurement, experiment, calculation or prediction.
What relevance, if any, does your CH-in-PRD hobbyhorse have for that
discussion?
Bill Taylor
|> In other words, if you like Occam's razor, you'll like CH.
Right on. However, as I've said several times, the same Occam will have you
refusing to accept AC. Occam's for ZF, not ZFC.
|> As always, should my comments prove to be way off base,
They're well on base; only you've got more bases to make yet!
|> just pretend I don't exist.
Personally, I believe in solipsism, but that's just one man's opinion.
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Kleeneness is next to Godelness.
-------------------------------------------------------------------------------
Bill Taylor
|> >Come on Dave - this won't do. One can't meaningfully compare theories which
|> >are INTENDED to have more than one model, like group theory, ring theory, etc,
|> >with those that are intended to have only one
|> Well, maybe you have a point, it's possible I was stretching here.
DAMME SIR! What's this, a retraction of sorts? Gad sir, this is usenet; one
is supposed to dig one's heels in and never ever retract anything, dontcha know?
That's damned unmanly, if not unAmerican activity! What are they doing there in
Oklahoma State? You'll be up before the HUUAC soon! (UnUsenetlike Activities)
|> I'm never quite certain what the "real" model of ZF is
Join the club.
|> (unless we're talking V=L,
AAAAAAAAAARRRRRRGGGHH!
|> Years ago it would have been clear to me what the real model was,
<snigger>
|> but some years before that it was
|> clear to everybody what the real model of naive (inconsistent) set theory was
<snort!> My very feelings on both counts.
The only more-or-less precise model(?) I know of, is the definable sets. That
is, sets definable in FOL with = and eps. Essentially, sets given by {x|P(x)}
terms. This collection is only countable of course, and thoroughly ineffectively
given, but is at least a precisely given collection. Of course it is not
definable itself within ZF, or any reasonable extension!
I'm not at all sure this collection actually *is* a model of ZF; I'm pretty
sure it's *not* a model of ZFC. But I'm really hoping some expert will
answer both of these questions for me; I'm out of my depth.
Can ayone help?
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
I don't like it, and I'm sorry I ever had anything to do with it. - Schrodinger
- but should have been Zermelo
-------------------------------------------------------------------------------
Ilias Kastanas
>>ika...@alumni.caltech.edu (Ilias Kastanas) wrote:
>
>[PA+CH is equivalent to PA omitted]
>
>>> So you don't really need CH etc., but there is no loss in
>>> assuming it if you want to; it might facilitate your argument.
>>> There is no mathematical need; it is psychological! You might of
>>> course annoy some people and incur their wrath...
>
>> Huh. I wonder if this ever happens with, say, examples in analysis.
>
>non-standard analysis (and there are results with no known standard
>proofs) often rely on "saturation" properties. These are much easier
>to derive assuming CH.
>
>"Weak enough" refers to the Shoenfield absoluteness theorem--a statement
>of the form (for all reals) (there exists a real) (... arbitrary integer
>quantification ...) has the same truth value between models with the same
>aleph_1.
>
>Did you know that there are continuous nowhere differentiable functions
>of fractal dimension 1? Did you know that most continous real-valued
>functions on [0,1]--in the sense of category--are of fractal dimension
>1? I believe this is original with me. What makes it interesting is
>that I first proved this using Cohen forcing.
>
>The category proofs, and even various specific constructions, are trivial
>modifications of the usual proofs for continuous nowhere differentiable
>functions. Once you know this result, it should take about five minutes
>to take such a proof and extend it to fractal dimension 1.
>
>But that was _not_ how I stumbled upon it. I had never heard of the result,
>and the impression I got from Mandelbrot and friends was that the graphs
>of these bad boy functions were the canonical examples of fractals. I was
>doodling at some physics lecture, drawing dots at random, when it occurred
>to me out of nowhere that the dots formed a forcing condition, whose generic
>"limit" is _obviously_ continuous but nowhere differentiable. (The stronger
>conditions will fill the dots in a "generic" manner, avoiding any pattern,
>ie, zigzaggy on every scale.)
>
>It didn't take much longer to notice that the generic "limit" had fractal
>dimension 1 (and that "most" of its Besicovitch subdimensions were zero).
>(The stronger conditions squeeze in arbitrarily close to any connect-the-dots
>interpolation.)
>
>A bit of careful fiddling was needed to see that the Schoenfield theorem
>applied, but it was very elementary. (It does not apply to the existence
>of nowhere differentiable functions, but it does to "uniformly" nowhere
>differentiable functions, ie, the necessary delta-epsilon failure can be
>quantified independently of abscissa.)
>
>And since forcing is, deep down, just the Baire category theorem, the
>existence of this approach isn't that surprising.
>
>What would be of real interest would be finding a Delta^1_3 property of
>the generic "limit", since that would bring large cardinals into play.
>
>> (Are there any "real" theorems of first-order PA the proof of which
>>is substantially easier given CH?)
>
>into a Diaphontine equation game, so they most certainly do exist.
Nice!
Nowhere-differentiable functions are quite interesting. If N is the
set of such functions in C[0,1], then N is a complete co-analytic ( i.e.
Pi-1-1 boldface) set, and so is the subset of N consisting of Besicovitch
functions, that is, functions that lack even one-sided derivatives, finite
or infinite.
Ilias
David Ullrich
>
>DAMME SIR! What's this, a retraction of sorts? Gad sir, this is usenet; one
>is supposed to dig one's heels in and never ever retract anything, dontcha know?
Didn't mean to offend. (Didn't really mean the retraction, so let me
take it back. There, feel better now?)
Timothy Chow
Ilias Kastanas <ika...@alumni.caltech.edu> wrote:
> On the topic of physical theories assuming CH, ~CH, and their
> significance:
>
> Suppose physical theories P and P' assume Euclidean and non-Euclidean
> geometry resp. Suppose one prevails. Does this warrant a conclusion
> about the "truth" or "falsity" of the Parallel Postulate -- in Mathematics?
No, of course not. This brings up an important point, which I should have
been clearer about earlier.
The way to get a physical experiment to test CH (akin to the experiment of
running a ZFC-engine to test Con(ZFC)) is *not* to formulate two physical
theories, one assuming CH and one assuming ~CH and deciding which one is
more successful. After all, one doesn't test the consistency of ZFC by
seeing which of ZFC+Con(ZFC) and ZFC+~Con(ZFC) yields a better physical
theory.
What one must *first* do is to convince oneself that there is something
out there that corresponds to our mental picture of the reals sufficiently
well that the question of the existence of a subset with certain properties
becomes a *physical* question. Then one must find an experiment whose
outcome depends on what the answer to the physical question is.
It might be argued that we can never get past the first part, of convincing
ourselves that there is something out there corresponding to our mental
picture of the reals. One argument might be that a finite number of
experiments, no matter how precise, could never sway us in one direction
or the other, because an infinite amount of information is needed to specify
the reals. This isn't a bad argument, but note that most of us have somehow
accepted the idea that the ZFC-engine tells us something about Con(ZFC),
i.e., we've convinced ourselves that there is an "out-there-ness" about
the integers, even though the same infinite-information objection applies.
This is where my comments about physics being more than "direct" experiments
come in.
It also might be argued that even if we did get past the first part, the
second part of finding a suitable experiment would not be possible. It's
kind of moot arguing about this point, though, unless we get past the first
part, in which case the battle is half over anyway.
Ilias Kastanas
Bill Taylor <w...@math.canterbury.ac.nz> wrote:
[ . . . ]
>|> (unless we're talking V=L,
>
>AAAAAAAAAARRRRRRGGGHH!
>
[ . . . ]
>The only more-or-less precise model(?) I know of, is the definable sets. That
>is, sets definable in FOL with = and eps. Essentially, sets given by {x|P(x)}
>terms. This collection is only countable of course, and thoroughly ineffectively
>given, but is at least a precisely given collection. Of course it is not
>definable itself within ZF, or any reasonable extension!
>
>I'm not at all sure this collection actually *is* a model of ZF; I'm pretty
>sure it's *not* a model of ZFC. But I'm really hoping some expert will
>answer both of these questions for me; I'm out of my depth.
>
>Can ayone help?
Well, I'll try!
From where do we obtain objects to play the role "value of x"? And
if 'a' is such an object, how do we evaluate P(a)? (forgive the mix of
syntax and semantics, for the sake of convenience).
It seems a structure is needed, with a binary relation, and the a's
will be elements of the structure... For this to make sense, it looks
like... horror... the structure already needs to be a model of set theory?!
How to escape? Can we think of P(x) as an "abstract" description of
a set? P and Q yield the same set if Ax (P(x) <=> Q(x)) is logically
valid (extensionality of sorts)... likewise, P is a subset of Q if Ax (P(x)
=> Q(x))... But what about membership? When is P a member of Q? The
idea '{P} subset of Q' has an obvious problem -- what formula could possibly
yield {P}? And in any case, why should these equivalence classes of
formulas ("Lindenbaum algebra"?) be set-like? We need to make the eps
relation satisfy appropriate axioms...
Then again, accepting an externally given structure defeats the
purpose. And if we bootstrap by starting with 0 and applying definability
to build successive layers, each one by P(x)'s interpreted in, and taking
parameters from, what has "already been defined", it is even worse: we are
(gasp!) in Satan's employ, building the arch-enemy, L!!! (Initial segments
of it, or the whole thing if we run through the "intuitive" ordinals, i.e.
get going... take union... get going... take union... ... ... take union...
get going... ).
As somebody said, "This is worse than being checkmated... this loses
my Queen!"
I don't know, Bill. Hobson's choice?!
Ilias
David Ullrich
ourselves that there is something out there corresponding to our mental
picture of the reals. One argument might be that a finite number of
experiments, no matter how precise, could never sway us in one direction
or the other, because an infinite amount of information is needed to specify
the reals.>
That might indeed be argued.
<This isn't a bad argument>
That's correct.
<but note that most of us have somehow
accepted the idea that the ZFC-engine tells us something about Con(ZFC),
i.e., we've convinced ourselves that there is an "out-there-ness" about
the integers, even though the same infinite-information objection applies.>
The two are not the same at all. We know exactly what it means to
say that ZFC is consistent: Write down all the theorems of ZFC one by one.
If you ever see a P appearing on the list with a "Not P" appearing elsewhere
on the list then ZFC is inconsistent, otherwise it's consistent. Nobody actually
knows whether ZFC is consistent or not but we all know what it means to say it
is.
You haven't given me a clue how tell whether the local structure of
the universe is R^4 or Q^4 or something else, even in principle. Maybe the idea
is you keep building bigger and better microscopes. But there are two big
problems: First, let's suppose for a second that you can build a microscope
with an arbitrary resolution (ie, given any desired resolution you can build
a microscope that resolves things at that sort of scale - I'm not stipulating
a microscope with infinitely fine resolution). No matter HOW fine a microscope
you build, Q and R are still going to look the same (as opposed to the situation
with ZFC: if it is inconsistent then the inconsistency will show up at some
point, we just don't know when.)
Also, unless quantum mechanics is simply wrong you simply can't build
microscopes with arbitrarily fine resolutions.
Wiener said something a while ago about people actually assuming CH
in quantum mechanics - he never answered Tal Kubo's question as far as I can see.
The question seemed to be roughly the same as the one I was curious about when
it came up (which I never got around to asing because was busy refuting his
statements about the fractal dimension of most continuous functions): Are these
guys actually assuming that CH is true "in the physical universe", whatever
that means, or is it just CH allowing them to do various formal things with
their Hilbert spaces?
Matthew P Wiener
>Wiener asked:
>>Did you know that there are continuous nowhere differentiable functions
>>of fractal dimension 1? Did you know that most continous real-valued
>>functions on [0,1]--in the sense of category--are of fractal dimension
>>1? I believe this is original with me. What makes it interesting is
>>that I first proved this using Cohen forcing.
> Well, today I'm "certain" that the graphs of "most" continuous
>functions have dimension 2.
>[David's proof omitted]
There is no contradiction between our proofs, since I was using "fractal
dimension" as a synonym for "Hausdorff dimension". David has told me via
e-mail that this is incorrect on my part. I think it merely dates me,
but I don't recall anymore.
I am certain his proof is correct, since I can rederive (rather trivially)
his result via the above mentioned forcing argument.
Timothy Chow
David Ullrich <ull...@math.okstate.edu> wrote:
> The two are not the same at all. We know exactly what it means to
>say that ZFC is consistent: Write down all the theorems of ZFC one by one.
>If you ever see a P appearing on the list with a "Not P" appearing elsewhere
>on the list then ZFC is inconsistent, otherwise it's consistent. Nobody
>actually knows whether ZFC is consistent or not but we all know what it
>means to say it is.
>
> You haven't given me a clue how tell whether the local structure of
>the universe is R^4 or Q^4 or something else, even in principle.
Well, I have trouble coming up with something *really* convincing because
I'm not completely convinced that it's possible (I'm also not convinced
that it's impossible). But to handwave a bit, let's consider a "butterfly
effect" scenario where we have a system where arbitrarily small perturbations
can cause arbitrarily large macroscopic effects. Suppose we model the system
using the real numbers, and by churning through some math we find that
certain macroscopic phenomena will appear in the system, but only if the
initial conditions are irrational numbers. Then if we observe the effects
actually happening, we might be inclined to think that the fine structure
of the universe "really is" more like R^4 than Q^4.
Rather far-fetched, I admit, but I don't think it's inconceivable.
David Ullrich
>Well, I have trouble coming up with something *really* convincing because
>I'm not completely convinced that it's possible (I'm also not convinced
>that it's impossible). But to handwave a bit, let's consider a "butterfly
>effect" scenario where we have a system where arbitrarily small perturbations
>can cause arbitrarily large macroscopic effects. Suppose we model the system
>using the real numbers, and by churning through some math we find that
>certain macroscopic phenomena will appear in the system, but only if the
>initial conditions are irrational numbers. Then if we observe the effects
>actually happening, we might be inclined to think that the fine structure
>of the universe "really is" more like R^4 than Q^4.
>
>Rather far-fetched, I admit, but I don't think it's inconceivable.
Well, far-fetched seems right. I tend to think that these chaotic
systems people talk about are only chaotic from a practical point of view,
not in an absolute sense.
Say we have a pencil we're trying to balance on its end, and due
to atmospheric effects we deduce that it will remain balanced forever if
and only if the original inclination is sqrt(2). Well, first I doubt that
we're going to know that - to know that the magic number is exactly sqrt(2)
we need to know the wind velocity precisely, and we never will. Second,
suppose that we do know the wind velocity precisely, so we know the magic
number is exactly sqrt(2). Now we balance the pencil. It's not going to
stand on end forever, because in fact we're not going to be able to balance
it at exactly the right angle. And finally, supposing we do balance it at
exactly an angle of sqrt(2), we can't verify that it's remaining upright
forever because we don't have enough time. For all practical purposes, knowing
that it remains upright for a really really long time if the initial slope
is really really close to sqrt(2) is enough, but we're not talking about
practical purposes here.
Let me save you some time: You're going to say you didn't have such
a simple-minded example in mind. Maybe we look at a dynamical system and we
note that the behavior is periodic if a certain initial parameter is rational
while the orbits are dense if the parameter is irrational. This does not give
us a way to tell whether the initial parameter is rational or not, because we
cannot tell from observing the system whether the behavior is really periodic
or not.
A trivial example, in mathematics: Let x_n be the fractional part of
n*x_1 for some x_1 between 0 and 1. Now it's easy to see that the x_n are dense
in [0,1] if and only if x_1 is irrational - if x_1 is rational then the sequence
x_n is periodic. But there's no way to be certain whether the sequence is
periodic or not just by observing finitely many terms to any arbitrary but
finite precision. For example if
x_1 = 1/10 + sqrt(2)/10000000000000000000000000000000000000
then x_1 is irrational and the sequence x_n is dense in [0,1], but that sequence
is going to appear to be periodic for a long time, unless we look at the terms
incredibly closely.
For a long time the sequence will appear periodic with period 10. Now
if we look at it long enough we notice that this is actually not right, the numbers
x_(n*10) are not actually constant, they're moving very slowly. But if we have any
sense we will not conclude at this point that the sequence is dense, we will
conclude that we were fooled once, and we really have no way to tell whether the
sequence is dense or is simply periodic with a very large period (and is "almost
dense" in the sense that it eventually vists any interval of length at least
1/10000000000, but is not actually dense.)
David Ullrich
>>Wiener asked:
>
>>>Did you know that there are continuous nowhere differentiable functions
>>>of fractal dimension 1? Did you know that most continous real-valued
>>>functions on [0,1]--in the sense of category--are of fractal dimension
>>>1? I believe this is original with me. What makes it interesting is
>>>that I first proved this using Cohen forcing.
>
>> Well, today I'm "certain" that the graphs of "most" continuous
>>functions have dimension 2.
>
>>[David's proof omitted]
>
>There is no contradiction between our proofs, since I was using "fractal
>dimension" as a synonym for "Hausdorff dimension". David has told me via
>e-mail that this is incorrect on my part. I think it merely dates me,
>but I don't recall anymore.
There is this thing guys call "fractal dimension" - I've seen it
defined in several places and the definition's always the same. It does
seem quite curious that the fractal dimension of the graphs of "most"
continuous functions is 2, while the hausdorff dimension is 1.
I used to think that the fractal dimension thing was a silly device
constructed for the benefit of wimps/retards who couldn't sort out the definition
of hausdorff dimension. That was until last year when I learned a result
characterizing the fractal dimension of a graph in terms of the function's
smoothness. Then I thought about proving that theorem the other day (it's
one of those thesis/preprint things that I never followed up). As of the
last few days don't think the theorem's right - I can prove half of it and
disprove the other half. I think.
>
>I am certain his proof is correct, since I can rederive (rather trivially)
>his result via the above mentioned forcing argument.
Teehee (no flame intended). Actually it's two teehee's:
Teehee_1 : You know the proof's correct because you know the result's
correct. Presumably a joke, and a darn good one.
Teehee_2 : I think we were brought up different - what I did seems to me
like a trivial application of Baire's theorem plus the triangle inequality. I
find it amusing that a person would find a proof by forcing more believable
(but you probably feel the opposite, and that's OK...)
Matthew P Wiener
>But to handwave a bit, let's consider a "butterfly effect" scenario
>where we have a system where arbitrarily small perturbations can
>cause arbitrarily large macroscopic effects. Suppose we model the
>system using the real numbers, and by churning through some math we
>find that certain macroscopic phenomena will appear in the system,
>but only if the initial conditions are irrational numbers. Then if
>we observe the effects actually happening, we might be inclined to
>think that the fine structure of the universe "really is" more like
>R^4 than Q^4.
>Rather far-fetched, I admit, but I don't think it's inconceivable.
Yes, it is conceivable. At least one ideal version has been published.
There's somebody down in Georgia, name escapes me at the moment, who
has identified what he claims is a counterexample to the correspondence
principle. Specifically, the classical and the quantum Arnold cat have
different chaotic behavior. Its status as a counterexample depends on
whether it makes physical sense to distinguish between the rationals and
the irrationals.
(It appeared in PHYSICA D, about 3-5 years ago.)
Then again, there are some wild and crazy string theorists, who have
decided that 26 dimensions isn't room enough for them, and who work
in C cross all the p-adics.
David Ullrich
>different chaotic behavior. Its status as a counterexample depends on
>whether it makes physical sense to distinguish between the rationals and
>the irrationals.
OTOH this doesn't help us distinguish between a universe with and
without irrationals unless we can distinguish _by_ _observation_ between a
truly chaotic Arnold and an Arnold that just appears chaotic for a while
because we're not observing closely enough or for a long enough period.
Ilias Kastanas
David Ullrich <ull...@math.okstate.edu> wrote:
[ . . . ]
> Teehee_1 : You know the proof's correct because you know the result's
>correct. Presumably a joke, and a darn good one.
>
> Teehee_2 : I think we were brought up different - what I did seems to me
>like a trivial application of Baire's theorem plus the triangle inequality. I
>find it amusing that a person would find a proof by forcing more believable
>(but you probably feel the opposite, and that's OK...)
Then again, forcing IS Baire Category...
Ilias
Ilias Kastanas
Timothy Chow <tc...@math.lsa.umich.edu> wrote:
>In article <47d095$h...@gap.cco.caltech.edu>,
>Ilias Kastanas <ika...@alumni.caltech.edu> wrote:
>> On the topic of physical theories assuming CH, ~CH, and their
>> significance:
>>
>> Suppose physical theories P and P' assume Euclidean and non-Euclidean
>> geometry resp. Suppose one prevails. Does this warrant a conclusion
>> about the "truth" or "falsity" of the Parallel Postulate -- in Mathematics?
>
>No, of course not. This brings up an important point, which I should have
>been clearer about earlier.
>
>The way to get a physical experiment to test CH (akin to the experiment of
>running a ZFC-engine to test Con(ZFC)) is *not* to formulate two physical
>theories, one assuming CH and one assuming ~CH and deciding which one is
>more successful. After all, one doesn't test the consistency of ZFC by
>seeing which of ZFC+Con(ZFC) and ZFC+~Con(ZFC) yields a better physical
>theory.
>
>What one must *first* do is to convince oneself that there is something
>out there that corresponds to our mental picture of the reals sufficiently
>well that the question of the existence of a subset with certain properties
>becomes a *physical* question. Then one must find an experiment whose
>outcome depends on what the answer to the physical question is.
>
>ourselves that there is something out there corresponding to our mental
>picture of the reals. One argument might be that a finite number of
>experiments, no matter how precise, could never sway us in one direction
>or the other, because an infinite amount of information is needed to specify
It is worse than that. There is no experiment that will produce
even ONE real. And even if you grant that, say, the speed of light is a
real, there is no way to generate arbitrarily close rational approximations
("digits"). Or for any other physical constant. Finding a way to push
the boundaries of accuracy, for any one, is non-obvious; a genuine
discovery.
>the reals. This isn't a bad argument, but note that most of us have somehow
>accepted the idea that the ZFC-engine tells us something about Con(ZFC),
>i.e., we've convinced ourselves that there is an "out-there-ness" about
>the integers, even though the same infinite-information objection applies.
No quarrel about the integers, arithmetical statements have a clear
meaning, Con(ZFC) included. But the ZFC-engine... If ZFC is inconsistent
the engine will prove it. If it is consistent, it will never prove a
contradiction, or Con(ZFC) or something. At any moment it will have
shown finitely many theorems of ZFC, out of the infinitely many...
Without more, this is hardly a reason for increased faith in Con(ZFC).
The clarity of the set of integers is not the same as their being
"out there". No problem for "two notebooks" or "184 sheets of paper"...
but how would you argue that the full set of integers "exists" in the
world... before worrying about rationals and reals?
Ilias
Matthew P Wiener
>>Specifically, the classical and the quantum Arnold cat have
>>different chaotic behavior. Its status as a counterexample depends on
>>whether it makes physical sense to distinguish between the rationals and
>>the irrationals.
> OTOH this doesn't help us distinguish between a universe with and
>without irrationals unless we can distinguish _by_ _observation_ between a
>truly chaotic Arnold and an Arnold that just appears chaotic for a while
>because we're not observing closely enough or for a long enough period.
This is a pointless objection. You could say that about anything.
David Ullrich
>>find it amusing that a person would find a proof by forcing more believable
>>(but you probably feel the opposite, and that's OK...)
>
>
> Then again, forcing IS Baire Category...
Well so I hear. But we're still doing a lot of logic/model theory
when all we need is the triangle inequality. (Not that it matters.)
David Ullrich
Not certain what you mean by this, exactly. Could be you're taking my
comments as a hyper-skeptical "everything might be wrong" argument - I guess
the main reason I conjecture that might be what you mean is that arguments
like that are in fact pointless (if we don't know anything about anything
then there's no reason to suppose there's an actual keyboard in front of me at
this instant, so these typing motions my hands are apparently making seem
kind of pointless.)
But that's not what I meant. It's clear to me and I think to most
people that it's possible to determine that a certain distance is _not_
exactly one mile by measurement, but it's impossible to determine that it
_is_ exactly one mile by measurement - I can say this without any sort of
commitment to extreme skepticism.
Like I said, I'm not sure what you meant by "you could say that
about anything", quite. Another conjecture. Say hypothetically we have
some system and we know that the orbits are periodic except for initial
conditions in some exceptional set of measure zero. It seems possible
that you're claiming that I "could say" that we cannot determine by
observing the system whether the initial conditions are exceptional
or not. I would in fact say that - it seems to me that in such a case
people don't try to determine whether the initial conditions were
exceptional by observing the orbits, they simply conclude that in the
real world the orbits _are_ dense.
Not sure what your objection to my objection was, exactly. Yes, I
could say what I said about any argument requiring measurements of physical
quantities with infinite precision (or over an infinitely long time) - so
what, I _would_ say that about any such argument. That doesn't mean I would
say that about _anything_.
Matthew P Wiener
>>> I find it amusing that a person would find a proof by forcing more
>>>believable (but you probably feel the opposite, and that's OK...)
>> Then again, forcing IS Baire Category...
> Well so I hear. But we're still doing a lot of logic/model theory
>when all we need is the triangle inequality. (Not that it matters.)
The point is, the "lot of logic/model theory" is all in the background.
The analysis is reduced to instant obviousness.
Sort of like the Liouville theorem proof of the fundamental theorem of
algebra. A lot is swept under an easy-to-use rug, even if the original
weaving was somewhat tricky.
And the forcing proof has an overall advantage: to identify the second
category result, just study the one function's properties. One does not
need to invent new proofs for each result.
David Ullrich
>>>> I find it amusing that a person would find a proof by forcing more
>>>>believable (but you probably feel the opposite, and that's OK...)
>
>>> Then again, forcing IS Baire Category...
>
>> Well so I hear. But we're still doing a lot of logic/model theory
>>when all we need is the triangle inequality. (Not that it matters.)
>
>The point is, the "lot of logic/model theory" is all in the background.
>The analysis is reduced to instant obviousness.
>
>Sort of like the Liouville theorem proof of the fundamental theorem of
>algebra. A lot is swept under an easy-to-use rug, even if the original
>weaving was somewhat tricky.
>
"direct" proof was not so trivial.
Not that it matters.
>And the forcing proof has an overall advantage: to identify the second
>category result, just study the one function's properties. One does not
>need to invent new proofs for each result.
Golly. I guess the only reason there are any open problems left
is that you guys haven't got around to cranking out the solutions yet,
huh? (Sorry...)
Matthew P Wiener
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) wrote:
>>> OTOH this doesn't help us distinguish between a universe with and
>>>without irrationals unless we can distinguish _by_ _observation_ between a
>>>truly chaotic Arnold and an Arnold that just appears chaotic for a while
>>>because we're not observing closely enough or for a long enough period.
>>This is a pointless objection. You could say that about anything.
> Not certain what you mean by this, exactly. [...]
> It's clear to me and I think to most people that it's possible
>to determine that a certain distance is _not_ exactly one mile by
>measurement, but it's impossible to determine that it _is_ exactly
>one mile by measurement - I can say this without any sort of
>commitment to extreme skepticism.
You as might as well say "you can't distinguish between whether a certain
distance is is truly one mile or just appears to be one mile since we're
not observing precisely enough".
A pointless objection.
Matthew P Wiener
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) wrote:
>>Sort of like the Liouville theorem proof of the fundamental theorem of
>>algebra. A lot is swept under an easy-to-use rug, even if the original
>>weaving was somewhat tricky.
> Well I suppose - you'd have a more compelling point if the
>"direct" proof was not so trivial.
Compared to the forcing proof, it is not so trivial.
>>And the forcing proof has an overall advantage: to identify the second
>>category result, just study the one function's properties. One does not
>>need to invent new proofs for each result.
> Golly. I guess the only reason there are any open problems left
>is that you guys haven't got around to cranking out the solutions yet,
>huh? (Sorry...)
Is something bothering you?
David Ullrich
>distance is is truly one mile or just appears to be one mile since we're
>not observing precisely enough".
Don't know if I might "just as well" say that, but yes, I might say that.
>A pointless objection.
I don't think so - if a physical theory is such that verifying it
requires an infinitely precise measurement then that's a theory we're never
going to verify, hence a pointless theory.
Actual physics doesn't work that way - we carry out our measurements
to a certain finite precision. It doesn't take infinite precision to verify
that relaitivity and QM are right and classical physics is wrong. Well, it
would take infinite precision to be _certain_ modern physics was right, but
finite precision suffices to show that the classical versions are wrong.
(And I hope that most physicists wouldn't say they were certain of the truth
of modern physics, just that it seemed incredibly likely.) As far as I can see
(and I haven't seen any arguments to the contrary) it will take infinite
precision to decide whether there are really real reals in the actual
physical universe and then whether they satisfy CH or not - it seems like
a pointless question, being inherently untestable.
And I don't think that your comments on chaotic this and that really
say anything about this, because I don't see how we can distinguish between
something that's really chaotic and something that just appears that way
on a certain time scale. I need a positive epsilon and a measurement (whether
that be a measurement of how chaotic something is or something else) such that
if CH is true in the universe [whatever that means!!!] then the measurement
comes out bigger than epsilon, while if CH is really false then the measurement
will come out negative. How does that experiment go? I don't believe such an
experiment exists, although I'm not going to attempt to prove it doesn't.
David Ullrich
>all electrons have the same charge or just appear that way on the scale
>we measure things.
There's a difference. Although no measurement will ever suffice
to show beyond any doubt that a given two electrons have exactly the same
charge, if they do in fact have different charges a sufficiently precise
measurement will show this. Hence a series of experiments detecting no
difference might be taken as evidence that they have the same charge -
it's a falsifiable claim at least.
OTOH if we have a sequence of numbers defined by some physical
process I don't see how any finite measurement could show conclusively
that the sequence was periodic, nor do I see how a finite amount of
measurement could show that the sequence was not periodic. Neither the
statement nor its negation is falsifiable.
>A cheap and utterly pointless objection.
Do you _really_ think it even makes sense to claim that there
are structures in the physical universe corresponding precisely to the
real numbers, and that this is something that could be known by mortals?
Matthew P Wiener
>>You as might as well say "you can't distinguish between whether a
>>certain distance is is truly one mile or just appears to be one mile
>>since we're not observing precisely enough".
> Don't know if I might "just as well" say that, but yes, I
>might say that.
>>A pointless objection.
> I don't think so - if a physical theory is such that verifying
>it requires an infinitely precise measurement then that's a theory
>we're never going to verify, hence a pointless theory.
Who says it "requires" it?
> Actual physics doesn't work that way
Right. So your bringing it up was pointless.
> And I don't think that your comments on chaotic this and that really
>say anything about this, because I don't see how we can distinguish between
>something that's really chaotic and something that just appears that way
>on a certain time scale.
You as might as well say you can't see how we distinguish between whether
all electrons have the same charge or just appear that way on the scale
we measure things.
A cheap and utterly pointless objection.
Matthew P Wiener
>>> Well I suppose - you'd have a more compelling point if the
>>>"direct" proof was not so trivial.
>>Compared to the forcing proof, it is not so trivial.
>Even if you include all the required prerequisites? I don't think we're
>going to settle this.
Oh. Well, if you mean from scratch to final product, well yes, the
forcing proof is longer. Much longer.
In that sense, it is settled.
>No, that was a tiny joke - a retard could have taken "one does not
>need to invent a new proof for each result" as an assertion that
>forcing provided proofs for anything whatever, automatically. Please
>don't bother explaining to me that that'e not what you meant, I know
>that. It was a teensy joke.
OK. I won't explain what I meant.
Matthew P Wiener
>>You as might as well say you can't see how we distinguish between whether
>>all electrons have the same charge or just appear that way on the scale
>>we measure things.
> There's a difference. Although no measurement will ever suffice
>to show beyond any doubt that a given two electrons have exactly the same
>charge, if they do in fact have different charges a sufficiently precise
>measurement will show this. Hence a series of experiments detecting no
>difference might be taken as evidence that they have the same charge -
>it's a falsifiable claim at least.
And a claim that a certain system is periodic and not chaotic is in the
same falsifiability situation.
> OTOH if we have a sequence of numbers defined by some physical
>process I don't see how any finite measurement could show conclusively
>that the sequence was periodic, nor do I see how a finite amount of
>measurement could show that the sequence was not periodic. Neither the
>statement nor its negation is falsifiable.
The question isn't about arbitrary periods, but about the one we think
is there.
>>A cheap and utterly pointless objection.
> Do you _really_ think it even makes sense to claim that there
>are structures in the physical universe corresponding precisely to the
>real numbers, and that this is something that could be known by mortals?
An entirely different question.
David Ullrich
>>>You as might as well say you can't see how we distinguish between whether
>>>all electrons have the same charge or just appear that way on the scale
>>>we measure things.
>
>> There's a difference. Although no measurement will ever suffice
>>to show beyond any doubt that a given two electrons have exactly the same
>>charge, if they do in fact have different charges a sufficiently precise
>>measurement will show this. Hence a series of experiments detecting no
>>difference might be taken as evidence that they have the same charge -
>>it's a falsifiable claim at least.
>
>And a claim that a certain system is periodic and not chaotic is in the
>same falsifiability situation.
>
>> OTOH if we have a sequence of numbers defined by some physical
>>process I don't see how any finite measurement could show conclusively
>>that the sequence was periodic, nor do I see how a finite amount of
>>measurement could show that the sequence was not periodic. Neither the
>>statement nor its negation is falsifiable.
>
>The question isn't about arbitrary periods, but about the one we think
>is there.
>
with a given period by observation. Don't recall that we ever defined "the
question" precisely.
>>>A cheap and utterly pointless objection.
>
>> Do you _really_ think it even makes sense to claim that there
>>are structures in the physical universe corresponding precisely to the
>>real numbers, and that this is something that could be known by mortals?
>
>An entirely different question.
It was the question that started all this, or so I thought. My
recollection is that someone wanted to know whether CH was true in the
universe, my reaction was that this made no sense until we knew we had real
reals (and real sets, etc) in the universe. If I copied the question
wrong again sorry.
>--
>-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
Tal Kubo
In article <47t662$4...@netnews.upenn.edu>
wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>
>Specifically, the classical and the quantum Arnold cat have different
>chaotic behavior. Its status as a counterexample depends on whether it
>makes physical sense to distinguish between the rationals and the
>irrationals.
Any number of simple physical experiments discriminate between rationals
and irrationals, e.g., stroboscopic photography of objects in uniform
rotational motion. In light of this, what point (if any) is there to
the outre example above?
Matthew P Wiener
>>> Do you _really_ think it even makes sense to claim that there
>>>are structures in the physical universe corresponding precisely to the
>>>real numbers, and that this is something that could be known by mortals?
>>An entirely different question.
> It was the question that started all this, or so I thought.
It may have started all this, but it mutated along the way to the
philosophical particulars of Joseph Ford's alleged chaos theoretic
counterexample to the correspondence principle.
Tal Kubo
>David Ullrich <ullrich@math writes:
>>>> Do you _really_ think it even makes sense to claim that there
>>>>are structures in the physical universe corresponding precisely to the
>>>>real numbers, and that this is something that could be known by mortals?
>
>>>An entirely different question.
You haven't answered the question. Was your CH-in-PRD reference merely
the usual obfuscation, or does it indicate some conceivable relation
between properties of the Real Reals and experimental physics?
>
>> It was the question that started all this, or so I thought.
>
>It may have started all this, but it mutated along the way to the
>philosophical particulars of Joseph Ford's alleged chaos theoretic
>counterexample to the correspondence principle.
Likewise, the question remains whether your reference to said outre
"counterexample" is obfuscatory, or in any way relevant to the discussion.
How does it bear on the issue of distinguishing rationals from irrationals,
in a way different from the easy direct experiments (e.g., roll one circle
around another) that do the same?
Matthew P Wiener
>In article <47t662$4...@netnews.upenn.edu>
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>>Specifically, the classical and the quantum Arnold cat have different
>>chaotic behavior. Its status as a counterexample depends on whether it
>>makes physical sense to distinguish between the rationals and the
>>irrationals.
>Any number of simple physical experiments discriminate between rationals
>and irrationals, e.g., stroboscopic photography of objects in uniform
>rotational motion.
Does that in fact distinguish, or is there a physical limit beyond which
such an experiment gives no conclusion?
> In light of this, what point (if any) is there to
>the outre example above?
The failure of the correspondence principle. I myself am dubious about
the distinction being physically possible, but even beyond that, I would
say that at best Ford had discovered an example of noncommuting limits.
David Ullrich
>>don't bother explaining to me that that'e not what you meant, I know
>>that. It was a teensy joke.
>OK. I won't explain what I meant.
Feel free if you have nothing else to do. What I should've said was
"Please note that you need not explain that that wasn't what you meant - I know
that wasn't what you meant, and I'm pretty sure I know what you did mean. It
was a joke."
Ilias Kastanas
Tal Kubo <ku...@germain.harvard.edu> wrote:
>
>In article <47t662$4...@netnews.upenn.edu>
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>>
>>Specifically, the classical and the quantum Arnold cat have different
>>chaotic behavior. Its status as a counterexample depends on whether it
>>makes physical sense to distinguish between the rationals and the
>>irrationals.
>
>Any number of simple physical experiments discriminate between rationals
>and irrationals, e.g., stroboscopic photography of objects in uniform
>the outre example above?
Details, please? What stroboscope can achieve this? Pardon my
simple-mindedness, but any physical experiment seems to have a certain
number of significant digits of accuracy. How do you distinguish an
arbitrary irrational from any of its rational approximations?
Ilias
Tal Kubo
Matthew P Wiener <wee...@sagi.wistar.upenn.edu> wrote:
>
>>>Specifically, the classical and the quantum Arnold cat have different
>>>chaotic behavior. Its status as a counterexample depends on whether it
>>>makes physical sense to distinguish between the rationals and the
>>>irrationals.
>
>>Any number of simple physical experiments discriminate between rationals
>>and irrationals, e.g., stroboscopic photography of objects in uniform
>>rotational motion.
>
>such an experiment gives no conclusion?
"A cheap and utterly pointless objection". "You could say that about
anything" -- including, as Ford notes in his paper, the QM failure he
claims to have found.
>> In light of this, what point (if any) is there to
>>the outre example above?
>
>The failure of the correspondence principle.
That was Ford's motivation, but is irrelevant for this discussion.
You cited his Physica D paper as going beyond garden-variety
commensurability effects, to the question of "whether it makes physical
sense to distinguish between the rationals and irrationals". As far as I
can tell, his paper doesn't touch on any aspect of this beyond the usual.
He merely bases his claims on commensurabilities having measure 0.
> I myself am dubious about
>the distinction being physically possible,
Presumably you are far more skeptical about the possibility of physical
tests of CH (AC, V=L, etc). As dubious, say, as with respect to the
possibility of evolution's Fact status being revoked sometime. Correct?
> but even beyond that, I would
>say that at best Ford had discovered an example of noncommuting limits.
The limits in question (large-time and small-hbar) were known not to
commute before then. If anything, it is hard to believe that they *could*
commute in general.
Matthew P Wiener
>In article <49i5ih$1...@netnews.upenn.edu>
>>David Ullrich <ullrich@math writes:
>>>>> Do you _really_ think it even makes sense to claim that there
>>>>>are structures in the physical universe corresponding precisely to the
>>>>>real numbers, and that this is something that could be known by mortals?
>>>>An entirely different question.
>You haven't answered the question.
I never claimed it did.
> Was your CH-in-PRD reference merely
>the usual obfuscation,
Go blow it out your rear. I was providing something for David Ullrich's
edification. What he makes of it is his business.
Anyway, good riddance.
Tal Kubo
>
>I never claimed it did.
"Never"? You've posted that PRD reference repeatedly, with assorted
claims attached. How silly of me to make the obvious inferences.
>
>> Was your CH-in-PRD reference merely
>>the usual obfuscation,
>
>Go blow it out your rear.
Charming. You get shriller and shriller in your old age.
> I was providing something for David Ullrich's
>edification.
How noble. I think he would have settled for an actual discussion.
> What he makes of it is his business.
Indeed.
Tal Kubo
>In article <49eivj$o...@decaxp.harvard.edu>,
>Tal Kubo <ku...@germain.harvard.edu> wrote:
>>
>>In article <47t662$4...@netnews.upenn.edu>
>>>chaotic behavior. Its status as a counterexample depends on whether it
>>>makes physical sense to distinguish between the rationals and the
>>>irrationals.
>>
>>Any number of simple physical experiments discriminate between rationals
>>and irrationals, e.g., stroboscopic photography of objects in uniform
>>the outre example above?
>
> Details, please? What stroboscope can achieve this? Pardon my
> simple-mindedness, but any physical experiment seems to have a certain
> number of significant digits of accuracy.
Please direct your question to Matthew. He has decreed that this line of
objection is "cheap and utterly pointless". So far as I can tell, the sort
of experiment I mentioned is no more vulnerable to that objection than the
alleged chaos theoretic QM failure.
> How do you distinguish an
> arbitrary irrational from any of its rational approximations?
I doubt it's physically possible. The ability to approximate within 1/N^2
using denominator at most N seems to rule it out: on any measuring scale
you need more accuracy to test commensurability than is available.
(Exponential sensitivity doesn't evade this, as (1+c/N^2)^N --> 1.)
Rational with denominator << experimental scale, is of course
experimentally viable.