Axiom of Choice
Nobuo Saito
I would like to know whether professional mathematicians believe
the Axiom of Choice or not.
I heard Goedel proved there will be no inconsistency
when the Axiom of Choice is added to the ZF set theory.
But I think this does not mean you can believe the Axiom of Choice.
Some people might say, you can build math on that axiom.
But from what I heard about Goedel, he believed the reality of
mathematics, for example, he believed the reality of real numbers,
so that he thought there must be the definite answer
to the continuum hypothesis.
As for me, I feel uncomfortable whenever I have to use the Axiom of
Choice,
except for a countable set.
Regards.
Robert Israel
Nobuo Saito <genki...@hotmail.com> wrote:
>I would like to know whether professional mathematicians believe
>the Axiom of Choice or not.
Yes, they do.
>As for me, I feel uncomfortable whenever I have to use the Axiom of
>Choice,
>except for a countable set.
Most mathematicians would feel no discomfort whatever in using the
Axiom of Choice. For example, functional analysts use the
Hahn-Banach Theorem without any qualms whatever.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Herman Rubin
Robert Israel <isr...@math.ubc.ca> wrote:
>In article <lwu1s3...@forum.mathforum.com>,
>Nobuo Saito <genki...@hotmail.com> wrote:
>>I would like to know whether professional mathematicians believe
>>the Axiom of Choice or not.
>Yes, they do.
Some do, and some do not.
>>As for me, I feel uncomfortable whenever I have to use the Axiom of
>>Choice,
>>except for a countable set.
>Most mathematicians would feel no discomfort whatever in using the
>Axiom of Choice. For example, functional analysts use the
>Hahn-Banach Theorem without any qualms whatever.
There is mathematics which requires some such proposition.
But a mathematician should almost always not use any
unnecessary axiom, and this includes the Axiom of Choice,
or any of its consequences.
An example of an unnecessary use of the nonconstructive
Hahn-Banach Theorem is in the necessary and sufficient
conditions for a solution to the Hamburger moment problem.
The existence of an explicit countable dense (in the
integral topology) set of continuous functions makes this
use of a non-constructive proposition unnecessary.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Stephen Montgomery-Smith
set theory) is consistent (no reason for this belief - just
a hunch).
I don't think that the set of real numbers has any independent
existence - they are merely a convenient abstraction.
I don't think there is enough reality to sets to make the
question of the truth of the axiom of choice meaningful.
Arguing this issue is about as sensible as arguing about
how many angels can fit on a pinhead, and future historians
will laugh at us for debating such things.
Nobuo Saito wrote:
>
> I would like to know whether professional mathematicians believe
> the Axiom of Choice or not.
>
> I heard Goedel proved there will be no inconsistency
> when the Axiom of Choice is added to the ZF set theory.
> But I think this does not mean you can believe the Axiom of Choice.
> Some people might say, you can build math on that axiom.
> But from what I heard about Goedel, he believed the reality of
> mathematics, for example, he believed the reality of real numbers,
> so that he thought there must be the definite answer
> to the continuum hypothesis.
>
> As for me, I feel uncomfortable whenever I have to use the Axiom of
> Choice,
> except for a countable set.
>
> Regards.
--
Stephen Montgomery-Smith
Department of Mathematics, University of Missouri, Columbia, MO 65211
Phone 573-882-4540, fax 573-882-1869
http://www.math.missouri.edu/~stephen ste...@math.missouri.edu
Hop David
Stephen Montgomery-Smith wrote:
> Myself, I don't even think ZFC (the usual axiom system for
> set theory) is consistent (no reason for this belief - just
> a hunch).
>
> I don't think that the set of real numbers has any independent
> existence - they are merely a convenient abstraction.
>
> I don't think there is enough reality to sets to make the
> question of the truth of the axiom of choice meaningful.
> Arguing this issue is about as sensible as arguing about
> how many angels can fit on a pinhead, and future historians
> will laugh at us for debating such things.
>
>
How many solid angles can fit on a pin head? If the head is a hemisphere
it's solid angle would be 2 pi r^2? And how many smaller solid angles
can that be partitioned into?
-- Hop
http://clowder.net/hop/index.html
Martin Goldstern
> I would like to know whether professional mathematicians believe
> the Axiom of Choice or not.
> ...
> As for me, I feel uncomfortable whenever I have to use the Axiom of Choice,
> except for a countable set.
I think that one of the "problems" with the axiom of choice is
that there are people (even mathematicians) who do not understand
it, in particular: do not understand that on many occasions (e.g.,
when dealing with a finite family, or with subsets of a fixed
countable or at least well-ordered set) an apparent use of AC
can be trivially eliminated.
Some are also not sure in what sense the axiom of choice, the
well-ordering theorem, and Zorn's lemma, are "equivalent".
Martin Goldstern
David C. Ullrich
an analyst who deals with these "real" numbers every day,
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote, in article
<3960DFF5...@math.missouri.edu>,
> Myself, I don't even think ZFC (the usual axiom system for
> set theory) is consistent (no reason for this belief - just
> a hunch).
Well, this isn't so clear to anyone else, but:
> I don't think that the set of real numbers has any independent
> existence - they are merely a convenient abstraction.
>
> I don't think there is enough reality to sets to make the
> question of the truth of the axiom of choice meaningful.
> Arguing this issue is about as sensible as arguing about
> how many angels can fit on a pinhead, and future historians
> will laugh at us for debating such things.
Good answer. Right on.
G. A. Edgar
> >the Axiom of Choice or not.
>
>
> Some do, and some do not.
I wonder if there is any mathematician who believes that the real
line is a countable union of countable sets.
--
Gerald A. Edgar ed...@math.ohio-state.edu
Keith Ramsay
hru...@odds.stat.purdue.edu (Herman Rubin) writes:
|In article <8jqj5r$h2a$1...@nntp.itservices.ubc.ca>,
|Robert Israel <isr...@math.ubc.ca> wrote:
|>In article <lwu1s3...@forum.mathforum.com>,
|>Nobuo Saito <genki...@hotmail.com> wrote:
|
|>>the Axiom of Choice or not.
|
|>Yes, they do.
|
|Some do, and some do not.
This question is complicated by the doubts about whether it has a
well-defined truth value. Some consider it akin to Euclid's fifth,
a proposition which can be chosen to be regarded as true or not as
we wish.
I haven't met many mathematicians who've considered it to have a
definite truth value (true or false) who think that it's false. There
are a few, but I think it's just a small few. I knew a mathematician
who apparently believed that the axiom of dependent choices was true,
and all sets of reals were measurable, and consequently the full axiom
of choice was false, although I don't know now whether this was just
his preferred arbitrary choice of axiom system, as opposed to what he
believed was definitely true of the real world of sets.
Practice is a somewhat independent question, however:
|>>As for me, I feel uncomfortable whenever I have to use the Axiom of
|>>Choice,
|>>except for a countable set.
|
|>Most mathematicians would feel no discomfort whatever in using the
|>Axiom of Choice. For example, functional analysts use the
|>Hahn-Banach Theorem without any qualms whatever.
It seems to me that mathematicians have an overwhelming preference for
proceeding to use the axiom of choice whenever the usual occasions for
its use arise.
|There is mathematics which requires some such proposition.
|But a mathematician should almost always not use any
|unnecessary axiom, and this includes the Axiom of Choice,
|or any of its consequences.
|
|An example of an unnecessary use of the nonconstructive
|Hahn-Banach Theorem is in the necessary and sufficient
|conditions for a solution to the Hamburger moment problem.
|The existence of an explicit countable dense (in the
|integral topology) set of continuous functions makes this
|use of a non-constructive proposition unnecessary.
Separability conditions often help in this respect, don't they?
There seems to be just enough caution applied to the axiom of choice
that mathematicians will *sometimes* note when it's being used. It
would be my preference that they always noted the need for it, and
tried to find a proof using only dependent choice if possible. But I
think the typical mathematician considers this the domain of the
logicians, and doesn't want to take the trouble to so much as learn
how to tell when choice is being used or not, let alone subtleties
like the difference between countable choice and dependent choice.
There's one obvious category of instances when mathematicians make
unnecessary use of an axiom, and that's the huge number of avoidable
uses of the law of excluded middle which they make. For mathematicians
to adopt for both the axiom of choice _and_ the law of excluded middle
the policy you propose for the axiom of choice would greatly please
constructivists. Doing it for only the axiom of choice is much closer
to actual current practice, but it's not clear to me whether it helps
very much.
Keith Ramsay
Herman Rubin
G. A. Edgar <ed...@math.ohio-state.edu.nospam> wrote:
>> >I would like to know whether professional mathematicians believe
>> >the Axiom of Choice or not.
>> >Yes, they do.
>> Some do, and some do not.
>I wonder if there is any mathematician who believes that the real
>line is a countable union of countable sets.
This is consistent within ZF.
Bill Taylor
|> >> >the Axiom of Choice or not.
|>
|> >> >Yes, they do.
|>
|> >> Some do, and some do not.
Correct. Though the vast majority do. I do not; though this is due to
having a quite different view od what sets actually *are* than most mathies.
|> >I wonder if there is any mathematician who believes that the real
|> >line is a countable union of countable sets.
Yes, me again. This is an excellent example question, BTW.
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Why is there a numerical difference between monologue and diatribe?
-------------------------------------------------------------------------------
Bill Taylor
|> >Choice, except for a countable set.
Join the club. Incidentally, you probably would like to extend your comfort
zone from mere countable AC, to DC - the axiom of dependent choices.
This is strictly stronger - it says not only can you choose one from each of
a countable collection, but can do it even if each choice may have to depend
on the previous ones. Most ctbl-AC-ers would have no difficulty with this
extension, which is far and away the most commonly used in analysis etc.
|> Most mathematicians would feel no discomfort whatever in using the
|> Axiom of Choice.
True; but quite a few do.
Incidentally, I was reading on the foundations-of-math mailing list recently,
that for quite some time theorems in journals would be stated with the proviso
"AC needed", when it was; and that this gradually faded away as authors (and
journals) gradually became inured to it. I hadn't been aware of this before,
I'd thought that as soon as it was identified (Zermelo 1904?) it became
immediately accepted. Can anyone enlighten us further on this matter?
BTW, Freidman strongly suggests that large cardinal axioms will gradually
assume the same status as their essential use becomes better known in
mainstream math, in particular, in "Boolean relation theory", his own
particular baby. It will be interesting to see. I hope I live so long!
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Truth decays into beauty, while beauty soon becomes merely charm. Charm ends up
as strangeness, and even that doesn't last - but up and down is for a lifetime.
-------------------------------------------------------------------------------
Jim Heckman
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> [...]
>
> Incidentally, I was reading on the foundations-of-math mailing list
> recently, that for quite some time theorems in journals would be
> stated with the proviso "AC needed", when it was; and that this
> gradually faded away as authors (and journals) gradually became
> inured to it.
This is consistent with the accounts given in a couple of intoductory
textbooks on Set Theory I've read.
> I hadn't been aware of this before, I'd thought that as soon as it
> was identified (Zermelo 1904?) it became immediately accepted. Can
> anyone enlighten us further on this matter?
At least one of my textbooks claims that, in fact, when Zermelo proved
the Well-Ordering Principle, mathematicians were so astonished that
they looked for holes in his proof, and found: AC.
BTW, in ZF+DC, what can be said about the cardinality of the real
line R? It must still be Pow(aleph_0) =? 2^{aleph_0}, mustn't it? I'm
sure I've read that it's possible to prove that R is uncountable using
only its order properties, as opposed to the way it's usually done
using its algebraic properties, via infinite binary series, etc. And
I think I've read it's possible to prove that Pow(c) (=? 2^c) > c for
any cardinality c, even without AC. Does the existence of uncountable
well-ordered sets, which I *know* can be shown without full AC, depend
on DC?
--
~~ Jim Heckman ~~
-- "As I understand it, your actions have ensured that you will never
see Daniel again." -- Larissa, a witch-woman of the Lowlands.
-- "*Everything* is mutable." -- Destruction of the Endless
Sent via Deja.com http://www.deja.com/
Before you buy.
Mike Oliver
> |> >I wonder if there is any mathematician who believes that the real
> |> >line is a countable union of countable sets.
>
> Yes, me again. This is an excellent example question, BTW.
What? In a post dated 11 minutes later than the one to
which I'm responding, you counsel Nobuo Saito to "extend [his]
comfort zone from mere countable AC, to DC". Now, naively I
took that as saying that you're happy with AC_omega and indeed
with DC. This also seems to jibe with views you've expressed in
the past.
But even AC_omega is enough to refute the idea that the reals
can be partitioned into countably many countable sets.
Posted and mailed.
G. A. Edgar
<mat...@math.canterbury.ac.nz> wrote:
>>I wonder if there is any mathematician who believes that the real
>>line is a countable union of countable sets.
>
> Yes, me again. This is an excellent example question, BTW.
This reminds me of another example question... Dedekind finite sets:
Is there a set A, not equivalent to any proper subset of A, but
also not equivalent to any initial segment {0,1,2,...,n-1} of
the natural numbers?
[According to AC, no such sets exist.]
Or this: set B has a subset with n elements for every
natural number n, but B does not contain an infinite sequence
of distinct elements.
Gary McGuire
>I don't think that the set of real numbers has any independent
>existence - they are merely a convenient abstraction.
Do you (or people who agree with you)
think that the integers have an
independent existence?
-Gary McGuire
the_grea...@my-deja.com
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
> Myself, I don't even think ZFC (the usual axiom system for
> set theory) is consistent (no reason for this belief - just
> a hunch).
Knock knock, puddin head. :-)
OK, since you're unwilling to disclose your reasons, let
me interject with one of my own.
----------------------------------------------------
Cantorians believe that N can be put into bijection
with the finite power set of N. Their proposed
bijection might look something like this:
N FP(N)
=== =====
1 <-> {7,11}
2 <-> {1,3,9021}
3 <-> {4}
4 <-> {1,2}
5 <-> {81,82,83}
. etc. .
Recall, a bijection is a "one-to-one match-up"
between the elements of two sets, with none left
over in either set.
I will now demonstrate, that, no such bijection
is possible between N and FP(N).
The bijection below pairs each element of N to a set
in FP(N) containing it (the equivalent element of N)
as one of its members.
N FP(N)
=== ===
1 <-> {1}
2 <-> {1,2}
3 <-> {1,2,3}
4 <-> {1,2,3,4}
5 <-> {1,2,3,4,5}
. etc. .
I will reach the bijection above, by using valid
Cantorian manipulations, performed on an assumed
valid starting bijection. Thus, any inconsistency
(contradiction) derived, from the bijection above,
invalidates either the starting bijection or the
presumed valid manipulation methods.
To convert the original bijection into the one
above, a simple step-by-step process, guaranteed
not to corrupt the bijection status, is used:
Starting with 1 and counting 1,2,3, etc. find the
first element of N that is not correctly (as shown
above) paired to a set in FP(N). Call this the nth
element of N. It should be paired to an n-element
set in FP(N), having 1 as its 1st element and n as
its n-th element. Once the correct set is located
(it has to be somewhere in FP(N)), swap the
incorrectly paired set, with the correct one.
Continue this process for all members of N.
Now, every member of N is paired, as shown above.
What member of N is {3,7} paired to? It has to be
paired, because of the assumed starting bijection.
But, it can't be paired, because of the ending
bijection. Contradiction! That proves N can not
be placed in bijection with FP(N) OR the Cantorian
methods used to manipulate the pairings are invalid.
Nathan the Great
Age 12
Stephen Montgomery-Smith
I don't think that the "set of integers" has any existence. I do
think that each integer - at least those we can express, like
1,000, or Ackerman(10,10) do have some kind of independent
existence (well, I'm not so sure about Ackerman(10,10) - it
is so incredibly big and who is ever going to be able to count that
far to check it out).
Certainly the principle of induction is to my mind a great leap of
faith.
But I must admit that I get fuzzy when dealing with such questions.
And these qualms certainly don't stop me using these concepts with
great liberality when doing math. (I guess that I have about the
same uneasy feelings about using axiom of choice as I do induction,
which is why I have no problem using axiom of choice when it is
convenient.)
I'm not sure about the people who agree with me, I don't meet many
of them, although I now find out that David Ullrich is one of them.
By the way - this Ackerman function I mention - I don't recall the
exact definition - it is some very clever recursive thing, but the
idea is:
Ackerman(x,0) = x+1
Ackerman(x,1) = x+x
Ackerman(x,2) = x*x
Ackerman(x,3) = x^x
Ackerman(x,4) = x power itself x times
Ackerman(x,5) = previous function done to itself x times
......
I know this is not exactly right, but you can see that Ackerman(10,10) is
so incredibly big that it kind of shakes your belief that no integer can
be infinite.
Herman Rubin
Jim Heckman <jhec...@my-deja.com> wrote:
>In article <8jujca$d07$2...@cantuc.canterbury.ac.nz>,
>mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
..................
>> I hadn't been aware of this before, I'd thought that as soon as it
>> was identified (Zermelo 1904?) it became immediately accepted. Can
>> anyone enlighten us further on this matter?
>At least one of my textbooks claims that, in fact, when Zermelo proved
>the Well-Ordering Principle, mathematicians were so astonished that
>they looked for holes in his proof, and found: AC.
This was not the first explicit statement of AC. Russell is
certainly earlier.
>BTW, in ZF+DC, what can be said about the cardinality of the real
>line R? It must still be Pow(aleph_0) =? 2^{aleph_0}, mustn't it?
This does not require anything more than ZF. Some of us,
including myself, have even posted an explicit 1-1
correspondence, using continued fractions.
I'm
>sure I've read that it's possible to prove that R is uncountable using
>only its order properties,
A complete dense linearly ordered set with no first or last
element and a countable dense subset is isomorphic to the
reals. Other conditions can be used.
as opposed to the way it's usually done
>using its algebraic properties, via infinite binary series, etc.
The proof is not that much different.
And
>I think I've read it's possible to prove that Pow(c) (=? 2^c) > c for
>any cardinality c, even without AC.
This is essentially the Russell paradox.
Does the existence of uncountable
>well-ordered sets, which I *know* can be shown without full AC, depend
>on DC?
It depends on nothing beyond ZF. The Hartogs function,
the set of all equivalence classes of well-ordered sets
no larger than a given set, is not as small as that set.
Herman Rubin
G. A. Edgar <ed...@math.ohio-state.edu.nospam> wrote:
The oldest (Fraenkel) model of the consistency of the negation
of the Axiom of Choice in ZFU starts with an infinite set of
individuals whose only subsets are finite or co-finite. This
can be lifted to ZF using Cohen models.
david_...@my-deja.com
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
> Gary McGuire wrote:
> >
> > Stephen Montgomery-Smith :
> > >I don't think that the set of real numbers has any independent
> > >existence - they are merely a convenient abstraction.
> >
> > Do you (or people who agree with you)
> > think that the integers have an
> > independent existence?
> >
>
> I don't think that the "set of integers" has any existence. I do
> think that each integer - at least those we can express, like
> 1,000, or Ackerman(10,10) do have some kind of independent
> existence (well, I'm not so sure about Ackerman(10,10) - it
> is so incredibly big and who is ever going to be able to count that
> far to check it out).
>
> Certainly the principle of induction is to my mind a great leap of
> faith.
>
> But I must admit that I get fuzzy when dealing with such questions.
> And these qualms certainly don't stop me using these concepts with
> great liberality when doing math. (I guess that I have about the
> same uneasy feelings about using axiom of choice as I do induction,
> which is why I have no problem using axiom of choice when it is
> convenient.)
>
> I'm not sure about the people who agree with me, I don't meet many
> of them, although I now find out that David Ullrich is one of them.
Actually what I tend to agree with is what you said yesterday,
not what you're saying today. When you say the reals are a
mathematical abstraction I tend to agree. But the integers and
the set of integers seem perfectly concrete to me, where it
seems to me things get a little fuzzy is with the _power set_
of the integers (aka the reals).
I'm not sure about the people who agree with me here either,
but I suspect there are a lot more people who think of R as an
unreal abstraction than who think of N that way. Until just now
I've never heard anyone, well anyone with any credibility at
all, express any sort of unease regarding induction in N.
> By the way - this Ackerman function I mention - I don't recall the
> exact definition - it is some very clever recursive thing, but the
> idea is:
> Ackerman(x,0) = x+1
> Ackerman(x,1) = x+x
> Ackerman(x,2) = x*x
> Ackerman(x,3) = x^x
> Ackerman(x,4) = x power itself x times
> Ackerman(x,5) = previous function done to itself x times
> ......
> I know this is not exactly right, but you can see that Ackerman(10,10)
is
> so incredibly big that it kind of shakes your belief that no integer
can
> be infinite.
Um.
> --
> Stephen Montgomery-Smith
> Department of Mathematics, University of Missouri, Columbia, MO 65211
> Phone 573-882-4540, fax 573-882-1869
> http://www.math.missouri.edu/~stephen ste...@math.missouri.edu
>
david_...@my-deja.com
the_grea...@my-deja.com wrote:
> In article <3960DFF5...@math.missouri.edu>,
The methods you've used here are invalid. But they
have nothing to do with actual set theory except in your
imagination. I mean really, someone said something the
other day about how you come up with new errors all the
time, but that's not the way it looks to me, it looks
like the same old errors over and over. This one we
talked about years ago.
Or maybe that was someone else. Anyway, the error
is that the limit of a sequence of bijections need
not be a bijection. You construct those bijections
one by one, great. Now after you finish _all_ the
infinitly many modifications to the original bijection
you no longer have a bijection. This does not contradict
anything, except for what you (perhaps) _think_ "valid
manipulations" are.
For anyone who's actually trying to understand these
things: The limit of a sequence of bijections simply need
not be a bijection. Nobosy ever said it should be. A
person can give a much simpler example illustrating
why it isn't:
Let F_n be the bijection that maps 1 to n, decreases
all the numbers between 2 and n - 1 by one, and leaves
everything else fixed. So f_1 is the identity:
1 2 3 4 5 6 ...
1 2 3 4 5 6 ...
And f_2 looks like this:
1 2 3 4 5 6 ...
2 1 3 4 5 6 ...
And f_10 looks like this:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
2 3 4 5 6 7 8 9 10 1 11 12 13 14 ...
You get the picture. Each f_n is a bijection. But
as n -> infinity these bijections approach a map
which is not a bijection:
1 2 3 4 5 6 ...
2 3 4 5 6 7 ...
Nothing gets mapped to 1, so this map is not a
bijection.
There's no reason this map _should_ be a
bijection. And for _exactly_ the same reason
there's no reason the map that Nathan constructs
above should be a bijection - the fact that it
isn't proves nothing (otoh the fact that Nathan
thinks this gives a contradiction in set theory
proves something about Nathan, it's not clear
what. But all this has been explained to him
many times.)
Dave the Not Totally Stupid
> Nathan the Great
> Age 12
>
Martin Vaeth
>|>>I would like to know whether professional mathematicians believe
>|>>the Axiom of Choice or not.
>
>This question is complicated by the doubts about whether it has a
>well-defined truth value.
It is not a question which can be decided by mathematicians alone.
>I knew a mathematician
>who apparently believed that the axiom of dependent choices was true,
>and all sets of reals were measurable, and consequently the full axiom
>of choice was false
This is also my guess, although I would tend to replace "are measurable" by
"have the Baire property" (which has similar but even somewhat stronger
consequences in functional analysis and which in contrast to measurability
is provable to be *consistent* with ZF+DC by results of Shelah).
>There's one obvious category of instances when mathematicians make
>unnecessary use of an axiom, and that's the huge number of avoidable
>uses of the law of excluded middle which they make. For mathematicians
>to adopt for both the axiom of choice _and_ the law of excluded middle
>the policy you propose for the axiom of choice would greatly please
>constructivists. Doing it for only the axiom of choice is much closer
>to actual current practice, but it's not clear to me whether it helps
>very much.
I do not think that the question whether "it helps" is so important in this
connection. The more important question is: Does mathematics reflect reality
*with* or *without* AC/excluded middle...? I.e. are the properties of a
mathematical model for a physical phenomenon true in reality?
If not, then either the model is wrong, or the mathematical axioms are
`false' (in the sense of: inappropriate for reality).
So the question whether AC or e.g. "Shelah's axiom" hold might be decided by
an experiment (provided one has no doubts about the model). Sadly, I do not
have enough knowledge in physics to think up such an experiment, and most of
the physicists I asked about either have no interest or not enough experience
with AC to fully understand its consequences.
Richard Carr
:Date: Wed, 05 Jul 2000 17:14:21 GMT
:From: david_...@my-deja.com
:Newsgroups: sci.math
:Subject: Re: Axiom of Choice
:
:In article <8jvj77$bf0$1...@nnrp1.deja.com>,
"between 2 and n" (just in case of objections)
:everything else fixed. So f_1 is the identity:
:
Dave Seaman
<the_grea...@my-deja.com> wrote:
>I will now demonstrate, that, no such bijection
>is possible between N and FP(N).
You have not demonstrated any such thing.
>The bijection below pairs each element of N to a set
>in FP(N) containing it (the equivalent element of N)
>as one of its members.
> N FP(N)
> === ===
> 1 <-> {1}
> 2 <-> {1,2}
> 3 <-> {1,2,3}
> 4 <-> {1,2,3,4}
> 5 <-> {1,2,3,4,5}
> . etc. .
This is not a bijection. There are lots of finite subsets of N that are
not in the range of this mapping.
>I will reach the bijection above, by using valid
>Cantorian manipulations, performed on an assumed
>valid starting bijection. Thus, any inconsistency
>(contradiction) derived, from the bijection above,
>invalidates either the starting bijection or the
>presumed valid manipulation methods.
Correct. Either the mapping you began with is not a bijection, or else
your "manipulations" converted a bijection into something that is not a
bijection, which shows the "manipulations" are not valid. Either way, it
is *your* argument, not the countability of FP(N), that fails.
--
Dave Seaman dse...@purdue.edu
Amnesty International calls for new trial for Mumia Abu-Jamal
<http://www.refuseandresist.org/mumia/021700amnesty.html>
Mike Oliver
> The more important question is: Does mathematics reflect reality
> *with* or *without* AC/excluded middle...? I.e. are the properties
> of a mathematical model for a physical phenomenon true in reality?
I have doubts that this question can be made precise enough to
bear on AC, though you're certainly welcome to show me how
it might be. To exhibit a lower limit on the subtlety you'd
have to use, consider the Banach-Tarski paradox for a second.
A decomposition of a solid ball into five pieces that can
be reassembled into *two* solid balls of the same radius as
the original, certainly seems non-physical.
Except that phyisically, as far as we know at least, there *aren't*
any solid balls. What seems to be a solid ball is a cloud of discrete
elementary particles. And no one has ever made a knife sharp enough
to cut into an electron.
And this is the basic problem that colors any attempt to think of
the real numbers as actually *representing* (rather than being
abstracted from) physical quantities. We know physical quantities
only to some bounded precision; if the universe were in fact
discrete, but at a scale *smaller* than that precision, how
would we ever find out?
Even if we could represent the physical universe entirely using
rationals with some bounded denominator, the reals are nonetheless
clearly useful for describing and making predictions about the
world -- they're just more *convenient* than trying to do everything
with some huge collection of discrete entities. And at some point,
as I've remarked before, convenience becomes necessity: If I want
to go to the moon it's convenient for me to have a rocket, even
though "in principle" I could just jump.
And for this reason it becomes useful to know how the reals behave,
even if they correspond to nothing physical at all but are merely
abstractions. In doing so, it's useful to be able to bring results
from one area of mathematics to bear on another. Set theory is
a great help in this regard, because it allows us to code mathematical
objects motivated by very diverse considerations, into a common
and intuitive framework.
So it seems to me that the question "is AC true or false?" is not
a question about physics, but rather a question of what hangs
together best with the mental software of set theory as operated
by human beings. And on that basis, I say, AC is true.
Stephen Montgomery-Smith
> Actually what I tend to agree with is what you said yesterday,
> not what you're saying today. When you say the reals are a
> mathematical abstraction I tend to agree. But the integers and
> the set of integers seem perfectly concrete to me, where it
> seems to me things get a little fuzzy is with the _power set_
> of the integers (aka the reals).
>
> I'm not sure about the people who agree with me here either,
> but I suspect there are a lot more people who think of R as an
> unreal abstraction than who think of N that way. Until just now
> I've never heard anyone, well anyone with any credibility at
> all, express any sort of unease regarding induction in N.
>
OK, I'm a bit weird in this respect. But let me explain.
These days we tend to do our math in the framework of first
order logic, and ZFC. Now all of this clever stuff has only
been around about 100 years. 200 years ago - well I'm not so
sure about my dates - who even considered this stuff? I mean
imagine telling Laplace that a function is a subset of the
cartesian product of two sets. Wouldn't he think you crazy?
Who is to say that in 200 years time that mathematics will have
advanced so far that our seemingly sophisticated ideas of
logic, and how we construct power sets of power sets of power sets
quotiented by equivalence relations is how we think of the
real numbers, will seem foolish and quaint. They will point
out obvious flaws (like we do to Euclid) that to us are totally
hidden. They will have a totally different axiom system - that
is if they still use axioms as the basis of what they do.
I think the greatest argument for the consisency of mathematics,
at least in some form, is that when we do the calculations, we
actually get the same results each time, and that the calculations
actually seem to work. We do thousands of calculations and
logical steps, and use them to try to get a man to the moon, and
amzazingly enough he actually gets there, and this happens so often
that we cease to be amazed. Or we balance our checkbooks, and we
are so sure that addition is associative and commutative that if
we get a different answer than our bank statement, we immediately
assume someone made a mistake - a mistake that could be corrected
with time - rather than perhaps this one time the usual laws of
arithmetic have failed. We just take this for granted, never
amazed at this.
But this argument, to my mind, becomes weaker when we take
abstraction to its limit. Certainly all this stuff about
large cardinals seems very dangerous to me. Any small
inconsistency that might be found in ZFC will probably kill
large cardinals stone dead. But probably the type of mathematics
that we do to get man to the moon will still survive.
This is not to say that I think that first order logic and
ZFC are very clever. I use them professionally all the time.
I think they are wonderful, to me they are tools to get by,
and very good tools they are. But maybe in a few years they
will be as quaint as the pieces of struck flint that we
dig up.
As I said, this is all very fuzzy stuff. We can all have differences
of opinions, and really noone can prove the other wrong.
Stephen
Torkel Franzen
> These days we tend to do our math in the framework of first
> order logic, and ZFC.
Not really. Mathematicians know very little about first order logic
and ZFC, and there is no reason why they should know more.
> I think the greatest argument for the consisency of mathematics,
> at least in some form, is that when we do the calculations, we
> actually get the same results each time, and that the calculations
> actually seem to work.
We do a lot of computations with small numbers, yes. But why should
this tell us anything about the consistency of wild theories about
utterly hypothetical natural numbers?
Stephen Montgomery-Smith
>
> Stephen Montgomery-Smith <ste...@math.missouri.edu> writes:
>
> > These days we tend to do our math in the framework of first
> > order logic, and ZFC.
>
> Not really. Mathematicians know very little about first order logic
> and ZFC, and there is no reason why they should know more.
I guess I agree with this, but I think that first order logic and
ZFC permeate the culture in which we do mathematics.
>
> > I think the greatest argument for the consisency of mathematics,
> > at least in some form, is that when we do the calculations, we
> > actually get the same results each time, and that the calculations
> > actually seem to work.
>
> We do a lot of computations with small numbers, yes. But why should
> this tell us anything about the consistency of wild theories about
> utterly hypothetical natural numbers?
Yes, I would agree with this observation.
Torkel Franzen
>I guess I agree with this, but I think that first order logic and
>ZFC permeate the culture in which we do mathematics.
Yes, it's true that ZFC has an important foundational role.
>Yes, I would agree with this observation.
So if we did in fact base our confidence in mathematics on
statistics, that confidence would be quite unjustified. But in
actuality we don't.
Bill Taylor
|> Bill Taylor wrote:
|> > |> >I wonder if there is any mathematician who believes that the real
|> > |> >line is a countable union of countable sets.
|> >
|> > Yes, me again. This is an excellent example question, BTW.
|>
|> which I'm responding, you counsel Nobuo Saito to "extend [his]
|> comfort zone from mere countable AC, to DC".
Yes I did. But note, I was counselling *him* from his own POV, not saying
what my own preferences on the matter were. No conflict there.
|> Now, naively I
|> took that as saying that you're happy with AC_omega and indeed with DC.
You, naive, Mike? Get away. You were mistaken but understandably so.
I'm not *totally* happy with AC/DC, but I am so in practice. All uses of it
in "everyday" math seem to be quite innocent and aceptable. To find an
example use where it leads to a "false" conclusion is quite hard, and seems
always to involve the use of ordinals up to the first non-cionstructive
ordinal. These do not occur in ordinary analysis and so forth.
|> This also seems to jibe with views you've expressed in the past.
I hope things are clearer now?
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Is aleph_69 a cardinal sin?
-------------------------------------------------------------------------------
Bill Taylor
|> This reminds me of another example question... Dedekind finite sets:
|> Is there a set A, not equivalent to any proper subset of A, but
|> also not equivalent to any initial segment {0,1,2,...,n-1} of
|> the natural numbers?
I think that needs only a very minor version of AC to deny. Namely,
"every infinite set has a countable subset". I think this is strictly
weaker than ctbl AC. It would be interesting to know if this can be "reversed"
in Freidman style; i.e. that the nonexistence of such sets as A above
implies the ctble subset statement.
|> Or this: set B has a subset with n elements for every
|> natural number n, but B does not contain an infinite sequence
|> of distinct elements.
Good question. On the face of it this appears to be impossible in ZF alone:
just string together some 1-element, 2-element, 3-element... subsets of B,
and remove duplications, and you've got it. Unfortunately, ctbl AC is being
used here to pick out one of each n-element subset collection. Tough.
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
MATH: the discovery, clarification and rigorous study of
precise relationships in number, pattern, and structure.
-------------------------------------------------------------------------------
Pertti Lounesto
> Torkel Franzen wrote:
>
> > We do a lot of computations with small numbers, yes.
> > But why should this tell us anything about the consistency
> > of wild theories about utterly hypothetical natural numbers?
>
> Yes, I would agree with this observation.
If natural numbers are hypothetical, it would be more honest
to rename them as hypothetical numbers. Thus, I suggest
that we adopt more descriptive names for our numbers
natural numbers -> hypothetical numbers,
real numbers -> utterly hypothetical numbers,
complex numbers -> complex utterly hypothetical numbers.
Torkel Franzen
> If natural numbers are hypothetical, it would be more honest
> to rename them as hypothetical numbers.
A natural number may be more or less hypothetical, just as a penguin
may be more or less hypothetical. It's all a matter of what hypotheses
we use. It is not really feasible or advisable to introduce an
ontological category of hypothetical penguins.
Martin Vaeth
>
>> The more important question is: Does mathematics reflect reality
>> *with* or *without* AC/excluded middle...? I.e. are the properties
>> of a mathematical model for a physical phenomenon true in reality?
>
>I have doubts that this question can be made precise enough to
>bear on AC, though you're certainly welcome to show me how
>it might be.
>To exhibit a lower limit on the subtlety you'd
>have to use, consider the Banach-Tarski paradox for a second.
>A decomposition of a solid ball [...]
The Banach-Tarski paradox evidently is much too simple for such a task:
The fundamental problem in this example is that the `mathematical model'
for a (physical) body surely is not a set in R^3.
Actually, I am more thinking of an experiment in a field like quantum mechanics
(see below) where physicists do not have so many doubts about the correctness
of the model (of course, they still may be mistaken, but you may never avoid
this problem in principle).
>And this is the basic problem that colors any attempt to think of
>the real numbers as actually *representing* (rather than being
>abstracted from) physical quantities.
Yes, as long as you want to interpret them ``too directly''.
>We know physical quantities only to some bounded precision; if the universe
>were in fact discrete, but at a scale *smaller* than that precision, how
>would we ever find out?
This reminds my of a paper I read some years ago (from a serious mathematician;
I forgot the name, but I can look it up, if someone is interested): In this
paper, it was suggested to replace the reals by the algebraic numbers (and the
corresponding topology) for physics which leads to some interesting
consequences. However, I forgot the details.
>Even if we could represent the physical universe entirely using
>rationals with some bounded denominator, the reals are nonetheless
>clearly useful for describing and making predictions about the
>world -- they're just more *convenient* than trying to do everything
>with some huge collection of discrete entities. And at some point,
>as I've remarked before, convenience becomes necessity [...]
Nobody thinks that this convenience is bad, *provided in can be justified*:
Just chosing a simpler model because it is simpler to calculate with, may be
convenient, but it may be false as well.
To give an example, consider the differential equation which describes the
swinging of a string or a drum: Actually, this should not be a differential
equation but a *difference* equation (i.e. for *finitely many* points, namely
the "centers" of the atoms (forget Heisenberg here, for a moment)).
However, the differential equation *is* a justified model, because one can
prove that for "sufficiently many" atoms the solution is *sufficiently near*
to the solution of the differential equation, no matter where the "centers"
of the atoms actually are (and now also with Heisenberg there is no problem).
However, in particular in quantum mechanics, physicists use models where there
is no such mathematical justification or some which can only be given e.g.
with the aid of AC (*in principle*).
The most promising example in my eyes is when they use "singular" functionals:
It follows from "Shelah's axiom" (which I have mentioned in my previous post)
that the dual space of l_\infty is l_1 (and also L_\infty^*=L_1) so that in
particular there is no singular functional on l_\infty (like
Hahn-Banach limits etc).
So I hope that some of the models in which they use such singular functionals
to obtain certain properties does not "match" reality.
However, actually I fear that many physicists then would just throw away this
model instead of blaming AC for the failure.
Other promising examples are justifications of models where for the
convergence of the "approximating discrete system" to the "continuous system"
(in the sense described above), one needs to apply the Hahn-Banach theorem in
a nonseparable space (and can otherwise not do without it); a collegue has
mentioned such examples when I discussed with him about the problem.
Herman Rubin
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
>> Actually what I tend to agree with is what you said yesterday,
>> not what you're saying today. When you say the reals are a
>> mathematical abstraction I tend to agree. But the integers and
>> the set of integers seem perfectly concrete to me, where it
>> seems to me things get a little fuzzy is with the _power set_
>> of the integers (aka the reals).
ALL concepts are abstract mental constructs. Understanding
an abstract concept is quite different from understanding
it as an abstraction.
It is even a mistake to talk about THE integers. There are
concepts with different meanings but the equal extensions.
The integers as finite cardinal numbers and as finite
ordinal numbers are isomorphic, but are they the same?
In some MODELS they coincide, but this is not necessary,
nor does it change the applicability.
>> I'm not sure about the people who agree with me here either,
>> but I suspect there are a lot more people who think of R as an
>> unreal abstraction than who think of N that way. Until just now
>> I've never heard anyone, well anyone with any credibility at
>> all, express any sort of unease regarding induction in N.
This might be a good characterization of credibility. :-)
Don't expect it of elementary and high school teachers. :-(
>OK, I'm a bit weird in this respect. But let me explain.
>These days we tend to do our math in the framework of first
>order logic, and ZFC. Now all of this clever stuff has only
>been around about 100 years. 200 years ago - well I'm not so
>sure about my dates - who even considered this stuff? I mean
>imagine telling Laplace that a function is a subset of the
>cartesian product of two sets. Wouldn't he think you crazy?
He would not think that a function, from the standpoint of
use, being a particular type of subset of "the" Cartesian
product of two sets is crazy. There is a good reason for
putting "the" in quotes; there is no unique object which
can be called the Cartesian product.
What is a function, in its fullest generality? It is
"something" which takes arguments and produces values. Two
functions are equal if they have exactly the same arguments
and, for each argument, produce exactly the same values.
So any function as a black box is identified by the subset
of the Cartesian product.
This does not mean that a function should be defined this
way; the von Neumann axiomatization of set theory used
"function" and "argument" as primitive objects.
>Who is to say that in 200 years time that mathematics will have
>advanced so far that our seemingly sophisticated ideas of
>logic, and how we construct power sets of power sets of power sets
>quotiented by equivalence relations is how we think of the
>real numbers, will seem foolish and quaint.
Again, DEFINING the real numbers is a mistake.
Characterizing them as the completion of the rationals is
what is needed, and the various constructions are proofs
that such exists. Again, I am misusing "the".
They will point
>out obvious flaws (like we do to Euclid) that to us are totally
>hidden.
Euclid did not have all the axioms present, partly because
he knew of no possibility of them not holding. It was not
flaws, but gaps. On the other hand, the early attempts at
formulation of logic, even when paradoxes have occurred,
were not thrown out, but carefully axiomatized to eliminate
the paradoxes (we hope). This will continue.
They will have a totally different axiom system - that
>is if they still use axioms as the basis of what they do.
There will be extensions to the logical systems, but they
will still prove theorems.
>I think the greatest argument for the consisency of mathematics,
>at least in some form, is that when we do the calculations, we
>actually get the same results each time, and that the calculations
>actually seem to work.
Some of the time they agree, and some of the time they do
not. Those of us with more than a nodding acquaintance
with numerical mathematics are more concerned than you
seem to think. It is only with computations with integers
and symbolic objects what we get the same results every
time by using different computational methods. That
doing exactly the same thing gets the same answers does
not demonstrate anything.
We do thousands of calculations and
>logical steps, and use them to try to get a man to the moon, and
>amzazingly enough he actually gets there, and this happens so often
>that we cease to be amazed.
The mathematical complexity of getting a man to the moon
is actually quite low; 19th century methods would have
worked quite well. The amazement is that the physical
universe can be well approximated by mathematical models;
this has been called "the unreasonable effectiveness of
mathematics". But it is not a statement about mathematics
itself; Euclid knew quite well that the actual world, from
which geometry gets its name, did not satisfy the axioms.
Or we balance our checkbooks, and we
>are so sure that addition is associative and commutative that if
>we get a different answer than our bank statement, we immediately
>assume someone made a mistake - a mistake that could be corrected
>with time - rather than perhaps this one time the usual laws of
>arithmetic have failed.
We do have proofs of this, which is why we KNOW.
We just take this for granted, never
>amazed at this.
I see people assuming that the outputs of computers are
correct quite frequently, when the computer is either
doing a problem removed from the real problem, or is
making enough round off errors to give inaccurate values.
>But this argument, to my mind, becomes weaker when we take
>abstraction to its limit. Certainly all this stuff about
>large cardinals seems very dangerous to me. Any small
>inconsistency that might be found in ZFC will probably kill
>large cardinals stone dead. But probably the type of mathematics
>that we do to get man to the moon will still survive.
If we find an inconsistency in ZF (the C makes no difference),
we will have to patch it. It is possible that we can find
large cardinals inconsistent, but not, if ZF is consistent,
that we can find them consistent; this violates Godel's
results.
>This is not to say that I think that first order logic and
>ZFC are very clever.
The introduction which led to this was that of the
arbitrary use of variables in the late 16th century. The
use of logic was made by all of the early mathematicians,
mostly by the quotation of theorems. Limit processes go
back to the Greeks; they were not systematically used until
the late 17th century. In the 18th century, there were
lots of computations to be done, and foundational paradoxes
did not arise often enough to get attention, but they were
starting. It was necessary to use variables for functions,
not just numbers and similar objects; this leads to other
questions. Also, there was a problem, appearing
occasionally, about what such things as continuity meant.
There were differential equations, such as y'=sqrt(y),
which had non-unique solutions. All of these led to
investigation of foundations. Euler made use of
non-converging series, which clearly gave problems, and
made attempts to reconcile the problems. But it was clear
that, while the computations seemed to work, did they, or
did they just come close enough that the facilities of the
time could not find the error.
The 19th century was the period in which these questions
were looked into. The real numbers, continuity, the
meaning of derivative, etc. But algebra had already
started, and now one gets into logic, and when Boole
found an actual error in Aristotle, the questions got
more interesting. The Greeks already had one logical
paradox which they had never resolved, and one had
to set logic straight to be able to apply logic to
mathematics. The paradoxes in logic, and in set theory
after it was introduced, were close to the surface,
and related to each other.
What we have is not what we want; we want more. This
is what we have been able to understand. We will get
more, but just as Euclid's proofs are relevant, with
only his interpretation of axioms as "self-evident
truths" dropped, we will get an essentially conservative
extension of what we have now.
I use them professionally all the time.
>I think they are wonderful, to me they are tools to get by,
>and very good tools they are. But maybe in a few years they
>will be as quaint as the pieces of struck flint that we
>dig up.
There will certainly be some progress in higher order logic.
>As I said, this is all very fuzzy stuff. We can all have differences
>of opinions, and really noone can prove the other wrong.
--
David C. Ullrich
traditional to make it clear who said what,
hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>In article <3963A040...@math.missouri.edu>,
>Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
>
>>> Actually what I tend to agree with is what you said yesterday,
>>> not what you're saying today. When you say the reals are a
>>> mathematical abstraction I tend to agree. But the integers and
>>> the set of integers seem perfectly concrete to me, where it
>>> seems to me things get a little fuzzy is with the _power set_
>>> of the integers (aka the reals).
Actually I [David Ullrich] wrote that bit, not Stephen.
>ALL concepts are abstract mental constructs. Understanding
>an abstract concept is quite different from understanding
>it as an abstraction.
>
>It is even a mistake to talk about THE integers. There are
>concepts with different meanings but the equal extensions.
>The integers as finite cardinal numbers and as finite
>ordinal numbers are isomorphic, but are they the same?
>In some MODELS they coincide, but this is not necessary,
>nor does it change the applicability.
Well of course the integers are a mental construct as well,
and various isomorphic models are not actually the same
thing. This seems sort of obvious - when a person refers to
the integers as being more real than the reals a person
_presumably_ means that it seems clear that two persons
mental constructs of the integers are likely isomorphic,
while it's not at all so clear that this is true of the reals.
(That's just supposed to be an explanation of something
like what someone presumably means by statements like
the one of mine you quoted, not an argument for the
truth of the statement.)
>>> I'm not sure about the people who agree with me here either,
>>> but I suspect there are a lot more people who think of R as an
>>> unreal abstraction than who think of N that way. Until just now
>>> I've never heard anyone, well anyone with any credibility at
>>> all, express any sort of unease regarding induction in N.
>
>This might be a good characterization of credibility. :-)
I would have said the same a few days ago. But
we just saw Stephen Montgomery-Smith express unease
with induction on N - he's _certainly_ a competent
mathematician, and (although this is less certain) I
don't _think_ he was just trolling.
>Don't expect it of elementary and high school teachers. :-(
Alas, I wouldn't dream of expecting that.
David C. Ullrich
<ste...@math.missouri.edu> wrote:
>>
>> Actually what I tend to agree with is what you said yesterday,
>> not what you're saying today. When you say the reals are a
>> mathematical abstraction I tend to agree. But the integers and
>> the set of integers seem perfectly concrete to me, where it
>> seems to me things get a little fuzzy is with the _power set_
>> of the integers (aka the reals).
>>
>> I'm not sure about the people who agree with me here either,
>> but I suspect there are a lot more people who think of R as an
>> unreal abstraction than who think of N that way. Until just now
>> I've never heard anyone, well anyone with any credibility at
>> all, express any sort of unease regarding induction in N.
>>
>
>OK, I'm a bit weird in this respect. But let me explain.
>
>These days we tend to do our math in the framework of first
>order logic, and ZFC. Now all of this clever stuff has only
>been around about 100 years. 200 years ago - well I'm not so
>sure about my dates - who even considered this stuff? I mean
>imagine telling Laplace that a function is a subset of the
>cartesian product of two sets. Wouldn't he think you crazy?
>
>Who is to say that in 200 years time that mathematics will have
>advanced so far that our seemingly sophisticated ideas of
>logic, and how we construct power sets of power sets of power sets
>quotiented by equivalence relations is how we think of the
>real numbers, will seem foolish and quaint. They will point
>out obvious flaws (like we do to Euclid) that to us are totally
>hidden. They will have a totally different axiom system - that
>is if they still use axioms as the basis of what they do.
Could be. Could be that natural numbers don't
come up anymore, although I don't see how that would be.
But I really really really doubt that our current notion of
"natural number" will be considered somehow inconsistent,
or that induction on N will be considered invalid.
Luckily there's no way to prove that sort of
statement so I don't have to. But I really don't see how
it could happen.
>I think the greatest argument for the consisency of mathematics,
>at least in some form, is that when we do the calculations, we
>actually get the same results each time, and that the calculations
>actually seem to work. We do thousands of calculations and
>logical steps, and use them to try to get a man to the moon, and
>amzazingly enough he actually gets there, and this happens so often
>that we cease to be amazed. Or we balance our checkbooks, and we
>are so sure that addition is associative and commutative that if
>we get a different answer than our bank statement, we immediately
>assume someone made a mistake - a mistake that could be corrected
>with time - rather than perhaps this one time the usual laws of
>arithmetic have failed. We just take this for granted, never
>amazed at this.
>
>But this argument, to my mind, becomes weaker when we take
>abstraction to its limit. Certainly all this stuff about
>large cardinals seems very dangerous to me. Any small
>inconsistency that might be found in ZFC will probably kill
>large cardinals stone dead. But probably the type of mathematics
>that we do to get man to the moon will still survive.
That last sentence is a paraphrase of exactly what I
was going to say about this. Finding an inconsistency in ZFC
is not at all the same thing as finding that addition of
natural numbers is not commutative and associative (or
that induction doesn't work.)
>This is not to say that I think that first order logic and
>ZFC are very clever.
You meant(?) to say "This is not to say that
I do not think that..." .
> I use them professionally all the time.
>I think they are wonderful, to me they are tools to get by,
>and very good tools they are. But maybe in a few years they
>will be as quaint as the pieces of struck flint that we
>dig up.
>
>As I said, this is all very fuzzy stuff. We can all have differences
>of opinions, and really noone can prove the other wrong.
>
>Stephen
Mike Oliver
Bill Taylor wrote:
>
> Mike Oliver <oli...@math.ucla.edu> writes:
> |> Now, naively I
> |> took that as saying that you're happy with AC_omega and indeed with DC.
>
> You, naive, Mike? Get away. You were mistaken but understandably so.
> I'm not *totally* happy with AC/DC, but I am so in practice. All uses of it
> in "everyday" math seem to be quite innocent and aceptable. To find an
> example use where it leads to a "false" conclusion is quite hard, and seems
> always to involve the use of ordinals up to the first non-cionstructive
> ordinal. These do not occur in ordinary analysis and so forth.
>
> I hope things are clearer now?
>
So if I've got this straight, you're counting the statement
"the reals are not a countable union of countable sets" as a
*false* conclusion from AC_omega?
I have to say this seems extraordinarily at odds with your general
preference for not conferring ontological status to things you
can't exhibit with some degree of explicitness. Any partition
of the reals into countably many countable sets must *not* give
you a uniform way of verifying that the sets are in fact countable,
so I can't see how it could be explicit at all. The only way
I've ever heard of of constructing models with such sets is
to start with *more* sets, then cleverly "throw away" sets which
would allow you to verify that the countable union has only
countably many elements (and is therefore not all the reals).
I would call these models clearly pathological.
Also, of course, ordinary analysis would be seriously damaged
if the reals could be so partitioned -- measure and category
would both go away, since the reals would be the union of
countably many null sets and also of countably many meager sets.
Can you explain just *why* you think there's such a partition
of the reals?
Posted and mailed.
Stephen Montgomery-Smith
>
> I would have said the same a few days ago. But
> we just saw Stephen Montgomery-Smith express unease
> with induction on N - he's _certainly_ a competent
> mathematician, and (although this is less certain) I
> don't _think_ he was just trolling.
>
No, I wasn't trolling, but I was expressing views that
lie at the edge of my thinking.
Gary McGuire
>I don't think that the "set of integers" has any existence. I do
>think that each integer - at least those we can express, like
>1,000, or Ackerman(10,10) do have some kind of independent
>existence (well, I'm not so sure about Ackerman(10,10) - it
>is so incredibly big and who is ever going to be able to count that
>far to check it out).
I think that most people feel like the integers must really exist,
because we are so used to them. But as you say, we are only
used to the small ones. On the basis of our
experience with these, we accept on faith that the laws
(such as associativity of addition) hold for all integers.
But there are integers that we do not have direct experience
of, nor will we ever have, such as integers with 10^100 digits.
Maybe there are other laws for these numbers.
We know that strange things can happen in large dimensional
space that don't happen in small dimensions, so why can't
different things happen for large integers?
We have no experience and no way of knowing.
In fact, the number 3 is an abstract concept.
When we try to explain what 3 is, we run into trouble
and it seems we are forced to use set theory (Russell
sorted this out).
I'm not sure that integers have an independent existence.
It's just that we are used to them.
Platonists believe that mathematical objects really exist,
somewhere, but perhaps when you are just so used to thinking
about something, it becomes "real" to you.
I think it was von Neumann who said: you don't understand
mathematics, you just get used to it.
>Torkel Franzen:
>> We do a lot of computations with small numbers, yes. But why
should
>> this tell us anything about the consistency of wild theories about
>> utterly hypothetical natural numbers?
>Yes, I would agree with this observation.
Me too.
I'd say there's more to come about the integers.
Stephen Montgomery-Smith :
> > These days we tend to do our math in the framework of first
> > order logic, and ZFC.
I'm not a logician, but we all use the least-upper-bound axiom:
"all nonempty subsets of the reals with an upper bound
have a least upper bound"
which I don't think belongs to first-order logic
because it quantifies over subsets.
-Gary McGuire
Mike Oliver
va...@mathematik.uni-wuerzburg.de wrote:
> Mike Oliver <oli...@math.ucla.edu> wrote:
> >Even if we could represent the physical universe entirely using
> >rationals with some bounded denominator, the reals are nonetheless
> >clearly useful for describing and making predictions about the
> >world -- they're just more *convenient* than trying to do everything
> >with some huge collection of discrete entities. And at some point,
> >as I've remarked before, convenience becomes necessity [...]
> Nobody thinks that this convenience is bad, *provided in can
> be justified*: Just chosing a simpler model because it is simpler to
> calculate with, may be convenient, but it may be false as well.
I am less interested in whether it can be justified than in whether
it works. In my view Popperian antijustificationism needs to
be extended to the mathematics as well as the physics. Trying to
give a *motivation* for a theory is one thing; demanding a
*justification* strikes me as part and parcel of a misguided effort
to find in mathematics a justified, aprioristic certainty that
is not, in truth, available in any field of human endeavor.
> However, in particular in quantum mechanics, physicists use models
> where there is no such mathematical justification or some which can
> only be given e.g. with the aid of AC (*in principle*).
> The most promising example in my eyes is when they use "singular"
> functionals:
> It follows from "Shelah's axiom" (which I have mentioned in my
previous
> post)
You didn't say what it is, though; would you mind stating it?
And also if you know whether it's consistent with (or perhaps
even follows from) ZF+AD+DC?
> that the dual space of l_\infty is l_1 (and also L_\infty^*=L_1)
> so that in particular there is no singular functional on l_\infty
(like
> Hahn-Banach limits etc).
Embarassingly, I could also use a definition for "singular functional".
I feel I ought to be able to figure these things out from what you've
provided, but I do have other work :-).
> So I hope that some of the models in which they use such singular
> functionals to obtain certain properties does not "match" reality.
> However, actually I fear that many physicists then would just throw
> away this model instead of blaming AC for the failure.
I'm a little confused here -- if there's no such functional, wouldn't
that mean automatically that you have to throw away the model, given
that the model is predicated upon it? Or are you saying that the model
predicts that the behavior of the world is genuinely sensitive to the
existence/nonexistence of such a functional?
In the latter case, perhaps on reexamination (and certainly I'm
speculating here) what you'll find is that what it really depends
upon is the existence of such a functional in some definability
class; say, L(R). Given large cardinals, L(R) satisfies AD+DC.
If such a situation came to pass, one might very reasonably take
the attitude that the "physically real" type-2 objects were the
ones in L(R), but that mathematically AC still held in V (so
as to be able to make the most effective use of the machinery
of set theory). Then when it came time to apply results to
the physical world, you'd just have to remember that only some
type-2 objects, namely the ones in L(R), were directly relevant.
Richard Carr
:
:I'm not a logician, but we all use the least-upper-bound axiom:
:"all nonempty subsets of the reals with an upper bound
:have a least upper bound"
:which I don't think belongs to first-order logic
:because it quantifies over subsets.
What about if you write for all x in P(R) if there is a y in x then there
is a z in R such that for all w in x, w<=z and for all t in R if for all w
in x w<=t then z<=t. Then you quantify over elements of P(R). Since
P(R) is a set, you get back to first-order logic. (This works whenever you
quantify over P(X) if X is a set in the model rather than a proper class,
I think.)
:-Gary McGuire
:
Ray Vickson
Dave Seaman wrote:
I have not checked all the details, but I think in "Nathan's integers" he may
be
correct. If memory serves, Nathan was aged 12 more than two years ago, so
if he is measuring his age in integer years, 12 is obviously the largest
integer in
his system.
Richard Carr
:Date: Thu, 06 Jul 2000 17:47:14 -0400
:From: Ray Vickson <rvic...@engmail.uwaterloo.ca>
:Newsgroups: sci.math
:Subject: Re: Axiom of Choice
:
:
:
:
Surprsingly not. I think 12 is in fact consistent with what he wrote
originally . Of course he has been inconsistent in the past, there was
the time when he claimed he was 12 rather than 11.94... (or was it 11
instead of 10.94...) because he thought it was more accurate or better
that way or something (he only mentioned this when his inaccuracy was
pointed out to him) but Nathan hasn't been posting for too long- it just
seems that way.
:
:>
:>
:> --
:
:
Dave Seaman
Ray Vickson <rvic...@engmail.uwaterloo.ca> wrote:
>I have not checked all the details, but I think in "Nathan's integers" he may
>be
>correct. If memory serves, Nathan was aged 12 more than two years ago, so
>if he is measuring his age in integer years, 12 is obviously the largest
>integer in
>his system.
Not quite. Nathan claimed to be 11 when he first showed up in November,
1998. When challenged some time later, he allowed as how he may have
fudged his age by a few weeks, since he was slightly short of his 11th
birthday when he first began posting.
Martin Vaeth
>previous post)
>
>You didn't say what it is, though; would you mind stating it?
>And also if you know whether it's consistent with (or perhaps
>even follows from) ZF+AD+DC?
the property of Baire (i.e. it differs from an open set only by a meagure set).
I used the (not official) name "Shelah's axiom" because Shelah has proved that
it is consistent with ZF+DC. I do not know about the connection with AD,
although I were surprised if it were not at least consistent.
>Embarassingly, I could also use a definition for "singular functional".
The "usual" definition is as follows (this is only a sketch; for details see
books about Banach lattices, e.g. Luxemburg/Zaanen: Riesz Spaces):
Let X be a Banach lattice. Then any functional in the order dual has a
unique decomposition into a "regular" part (for l_\infty this corresponds to
a functional of l_1) and into a remainder which is the mentioned singular
functional.
>> So I hope that some of the models in which they use such singular
>> functionals to obtain certain properties does not "match" reality.
>> However, actually I fear that many physicists then would just throw
>> away this model instead of blaming AC for the failure.
>
>I'm a little confused here -- if there's no such functional, wouldn't
>that mean automatically that you have to throw away the model, given
>that the model is predicated upon it? Or are you saying that the model
>predicts that the behavior of the world is genuinely sensitive to the
>existence/nonexistence of such a functional?
I mean the latter.
Well, either they have to throw away the model or the underlying mathematical
logic (say: AC). What I am trying to say is that most physicist would then
not have the courage to blame AC.
>In the latter case, perhaps on reexamination (and certainly I'm
>speculating here) what you'll find is that what it really depends
>upon is the existence of such a functional in some definability
>class; say, L(R). Given large cardinals, L(R) satisfies AD+DC.
>
>If such a situation came to pass, one might very reasonably take
>the attitude that the "physically real" type-2 objects were the
>ones in L(R), but that mathematically AC still held in V (so
>as to be able to make the most effective use of the machinery
>of set theory). Then when it came time to apply results to
>the physical world, you'd just have to remember that only some
>type-2 objects, namely the ones in L(R), were directly relevant.
Yes, I completely agree with this. However, if it turns out that the only
mathematical objects relevant for physics are the constructible ones,
it would probably be more *convenient* to just throw AC overboard
(actually, this is what I mean when I say that AC does not hold).
Chan-Ho Suh
" I'm sure I've read that it's possible to prove that R is uncountable
using only its order properties, as opposed to the way it's usually
done using its algebraic properties, via infinite binary series, etc."
Yes, I've seen a proof on R's uncountability relying only on its
properties as a linear continuum. The general theorem says: An
infinite compact Haussdorf space with every pt. a limit pt. is
uncountable. I think.
Then you show an interval in R, i.e [0,1](in the order topology) to be
such a space.
The proof (as I recall) is in Munkres--Topology: A first course, if
you're interested.
The proof, to me, is very interesting because it uses a different
approach than the now old diagonalization argument.
Nathan the Great
a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
> >I will reach the bijection above, by using valid
> >Cantorian manipulations, performed on an assumed
> >valid starting bijection. Thus, any inconsistency
> >(contradiction) derived, from the bijection above,
> >invalidates either the starting bijection or the
> >presumed valid manipulation methods.
>
> Correct. Either the mapping you began with is not a
> bijection, or else your "manipulations" converted a
> bijection into something that is not a bijection, which
> shows the "manipulations" are not valid. Either way,
> it is *your* argument, not the countability of FP(N),
> that fails.
Dave, my step-by-step process can't corrupt the
bijection. It is an invariant transformation.
The bijection status is the invariant (unchanged)
property. And, regardless of the number of times
an invariant transformation is applied, invariant
properties do not change.
Cantor: "for sufficiently large 0's, 0+0 = oo"
--
Nathan the Great
Age 12
Nathan the Great
david_...@my-deja.com wrote:
> In article <8jvj77$bf0$1...@nnrp1.deja.com>,
> the_grea...@my-deja.com wrote:
> > Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
> > > Myself, I don't even think ZFC (the usual axiom system for
> > > set theory) is consistent (no reason for this belief - just
> > > a hunch).
> >
> > Knock knock, puddin head. :-)
> >
> > OK, since you're unwilling to disclose your reasons, let
> > me interject with one of my own.
> >
> > ----------------------------------------------------
> >
> > Cantorians believe that N can be put into bijection
> > with the finite power set of N. Their proposed
> > bijection might look something like this:
> >
> > N FP(N)
> > === =====
> > 1 <-> {7,11}
> > 2 <-> {1,3,9021}
> > 3 <-> {4}
> > 4 <-> {1,2}
> > 5 <-> {81,82,83}
> > . etc. .
> >
> > Recall, a bijection is a "one-to-one match-up"
> > between the elements of two sets, with none left
> > over in either set.
> >
> > I will now demonstrate, that, no such bijection
> > is possible between N and FP(N).
> >
> > in FP(N) containing it (the equivalent element of N)
> > as one of its members.
> >
> > N FP(N)
> > === ===
> > 1 <-> {1}
> > 2 <-> {1,2}
> > 3 <-> {1,2,3}
> > 4 <-> {1,2,3,4}
> > 5 <-> {1,2,3,4,5}
> > . etc. .
> >
> > Cantorian manipulations, performed on an assumed
> > valid starting bijection. Thus, any inconsistency
> > (contradiction) derived, from the bijection above,
> > invalidates either the starting bijection or the
> > presumed valid manipulation methods.
> >
> > above, a simple step-by-step process, guaranteed
> > not to corrupt the bijection status, is used:
> >
> > Starting with 1 and counting 1,2,3, etc. find the
> > first element of N that is not correctly (as shown
> > above) paired to a set in FP(N). Call this the nth
> > element of N. It should be paired to an n-element
> > set in FP(N), having 1 as its 1st element and n as
> > its n-th element. Once the correct set is located
> > (it has to be somewhere in FP(N)), swap the
> > incorrectly paired set, with the correct one.
> > Continue this process for all members of N.
> >
> > Now, every member of N is paired, as shown above.
> >
> > What member of N is {3,7} paired to? It has to be
> > paired, because of the assumed starting bijection.
> > But, it can't be paired, because of the ending
> > bijection. Contradiction! That proves N can not
> > be placed in bijection with FP(N) OR the Cantorian
> > methods used to manipulate the pairings are invalid.
>
> The methods you've used here are invalid. But they
> have nothing to do with actual set theory except in your
> imagination. I mean really, someone said something the
> other day about how you come up with new errors all the
> time, but that's not the way it looks to me, it looks
> like the same old errors over and over. This one we
> talked about years ago.
Stop repeating your errors, and I'll stop
repeating my corrections.
> Or maybe that was someone else. Anyway, the error
> is that the limit of a sequence of bijections need
> not be a bijection. You construct those bijections
> one by one, great. Now after you finish _all_ the
> infinitly many modifications to the original bijection
> you no longer have a bijection. This does not contradict
> anything, except for what you (perhaps) _think_ "valid
> manipulations" are.
Dave, those manipulations can't corrupt the
bijection. They are invariant transformations
of it. The bijection _status_ is an invariant
(unchanging) property. And, regardless of the
number of times an invariant transformation
is applied, invariant properties don't change.
> For anyone who's actually trying to understand these
> things: The limit of a sequence of bijections simply need
> not be a bijection. Nobosy ever said it should be. A
> person can give a much simpler example illustrating
> why it isn't:
That reminds me of a comment Cantor made while
being held captive at an insane asylum in germany:
"Just because 0+0 equals zero, doesn't mean that
0+0+0+... equals zero. In fact, I can prove
0+0+0+... equals whatever I want it to."
So, for anyone who's actually trying to understand
these things, consider: Cantorians _still_ believe that!
> Let F_n be the bijection that maps 1 to n, decreases
> all the numbers between 2 and n - 1 by one, and leaves
> everything else fixed. So f_1 is the identity:
>
> 1 2 3 4 5 6 ...
> 1 2 3 4 5 6 ...
>
> And f_2 looks like this:
>
> 1 2 3 4 5 6 ...
> 2 1 3 4 5 6 ...
>
> And f_10 looks like this:
>
> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
> 2 3 4 5 6 7 8 9 10 1 11 12 13 14 ...
>
> You get the picture. Each f_n is a bijection. But
> as n -> infinity these bijections approach a map
> which is not a bijection:
>
> 1 2 3 4 5 6 ...
> 2 3 4 5 6 7 ...
>
> Nothing gets mapped to 1, so this map is not a
> bijection.
>
> There's no reason this map _should_ be a
> bijection.
Did you spell "shouldn't" wrong? Or are
you sticking your head in the sand again?
> And for _exactly_ the same reason
> there's no reason the map that Nathan constructs
> above should be a bijection - the fact that it
> isn't proves nothing (otoh the fact that Nathan
> thinks this gives a contradiction in set theory
> proves something about Nathan, it's not clear
> what. But all this has been explained to him
> many times.)
>
> Dave the Not Totally Stupid
True, you're one iota short.
Nathan the Great
"G. A. Edgar" <ed...@math.ohio-state.edu.nospam> wrote:
> This reminds me of another example question...
> natural number n, but B does not contain an infinite sequence
> of distinct elements.
It reminds me of the often asked question:
Considering that the set of all natural numbers is static
and infinite, what finite natural number has an infinite
number of natural numbers less than it?
Mike Oliver
Martin Vaeth wrote:
> [Mike Oliver wrote:]
>> You didn't say what ["Shelah's axiom"] is, though; would you mind
>> stating it? And also if you know whether it's consistent with (or
>> perhaps even follows from) ZF+AD+DC?
> It is the statement that any subset of a complete separable metric space has
> the property of Baire (i.e. it differs from an open set only by a meagure set).
> I used the (not official) name "Shelah's axiom" because Shelah has proved that
> it is consistent with ZF+DC. I do not know about the connection with AD,
> although I were surprised if it were not at least consistent.
This follows from AD; it's just the determinacy of every Banach-Mazur
game. To play the Banach-Mazur game for a set A, first I choose a basic
open set, then you choose a basic open set inside mine, then I choose one
inside yours, and so on. One of us (doesn't matter which) is responsible
for making sure the intersection of the sequence is a singleton. Then I win
if the point in the intersection is an element of A; otherwise you win.
You have a winning strategy if and only if A is meager.
I have a winning strategy if and only if A is comeager in some
neighborhood. Determinacy means every A has one of these two
properties; it's not hard to recover that every set of reals
has the p.o.B.
>> In the latter case, perhaps on reexamination (and certainly I'm
>> speculating here) what you'll find is that what it really depends
>> upon is the existence of such a functional in some definability
>> class; say, L(R). Given large cardinals, L(R) satisfies AD+DC.
>>
>> If such a situation came to pass, one might very reasonably take
>> the attitude that the "physically real" type-2 objects were the
>> ones in L(R), but that mathematically AC still held in V (so
>> as to be able to make the most effective use of the machinery
>> of set theory). Then when it came time to apply results to
>> the physical world, you'd just have to remember that only some
>> type-2 objects, namely the ones in L(R), were directly relevant.
> Yes, I completely agree with this. However, if it turns out that the only
> mathematical objects relevant for physics are the constructible ones,
> it would probably be more *convenient* to just throw AC overboard
> (actually, this is what I mean when I say that AC does not hold).
Not at all, mon vieux -- this is the whole point! Considerations of
the wider universe V give us insight into the behavior of L(R) which
would be difficult otherwise to come by. For example, the large
cardinals in V give us that L(R) satisfies AD.
Because L(R) sits inside V in such a nice, stable way (it's a transitive
subclass, there's a simple definition with good absoluteness properties,
etc), there's very little cost associated with finding out about
L(R) by reasoning with V but restricting quantifiers when necessary,
rather than refusing ever to consider sets outside L(R).
Dave Seaman
Nathan the Great <the_grea...@my-deja.com> wrote:
>In article <8jvsca$r...@seaman.cc.purdue.edu>,
> a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
>> >Cantorian manipulations, performed on an assumed
>> >valid starting bijection. Thus, any inconsistency
>> >(contradiction) derived, from the bijection above,
>> >invalidates either the starting bijection or the
>> >presumed valid manipulation methods.
>> Correct. Either the mapping you began with is not a
>> bijection, or else your "manipulations" converted a
>> bijection into something that is not a bijection, which
>> shows the "manipulations" are not valid. Either way,
>> it is *your* argument, not the countability of FP(N),
>> that fails.
>Dave, my step-by-step process can't corrupt the
>bijection. It is an invariant transformation.
Seems to me we both agree that your process *does* corrupt the bijection.
However, we'll leave that aside for now.
Each transposition is a transformation that preserves bijections.
Without much trouble, you can show that the composition of any two such
transpositions is a transformation that preserves bijections. Likewise
for the composition of any three, and so on.
Exactly how for does that "and so on" apply? In other words, if we let
P(n) represent the statement "every composition of n transpositions
yields a transformation that preserves bijections," then we can ask for
which values of n is P(n) true. With a little thought, we can show:
1. P(1).
2. P(n) -> P(n+1) for all n.
Using mathematical induction, we draw the conclusion: P(n) holds for all
__________ n. I have left a blank in that sentence for you to fill in.
This is a test of your understanding of mathematical induction.
Ed Hook
cs...@cornell.edu (Chan-Ho Suh) writes:
|> Jim Heckman wrote:
|>
|> " I'm sure I've read that it's possible to prove that R is uncountable
|> using only its order properties, as opposed to the way it's usually
|> done using its algebraic properties, via infinite binary series, etc."
|> Yes, I've seen a proof on R's uncountability relying only on its
|> properties as a linear continuum. The general theorem says: An
|> infinite compact Haussdorf space with every pt. a limit pt. is
|> uncountable. I think.
More generally, if X is a complete metric space
and P is a nonempty perfect subset of X, then
P is uncountable.
|> Then you show an interval in R, i.e [0,1](in the order topology) to be
|> such a space.
|> The proof (as I recall) is in Munkres--Topology: A first course, if
|> you're interested.
To prove the above theorem, you can construct
an injection of the sequence space 2^|N into
P -- since the usual Cantor set is uncountable,
that proves it. (Of course, the original poster
might want to object at this point, since the
proof that the Cantor set is uncountable probably
uses that 2^\aleph_0 = c in some way ... ) The
"construction" depends on two results:
(1) if p is a limit point of A \subseteq X, where
X is a Hausdorff space, then every neighborhood of
p contains infinitely many points of A
(2) If { A_n }_{n > 0} is a nested sequence of
nonempty closed sets in the complete metric
space X with the property that diam(A_n) --> 0
as n --> \infinity, then \bigcap_{n > 0} A_n
is a singleton.
Then P is infinite (by (1)), so you can pick distinct
points p_0, p_1 in P. Let eps_1 = min(1/2, d(p_0,p_1)/3)
and define P(i) = { x \in P | d(x,p_i) <= eps_1 } for i=0, 1.
Then P(0), P(1) are disjoint closed sets with diameter
<= 1 ... and each contains infinitely many points.
Now you proceed inductively -- if you've managed to
define pairwise-disjoint closed sets P(a_1,a_2,...,a_n)
with diameter <= 1/n for every n-tuple of 0,s and 1's
with each set containing infinitely many points of P,
just imitate the above procedure "inside" each of these
sets. That is, given P(a_1,a_2,...,a_n), pick distinct points
p_{a_1,...,a_n,0} and p_{a_1,...,a_n,1} in there
(different from any points that may have been chosen at
an earlier stage -- probably unnecessary) and define
P(a_1,a_2,...,a_n,i) for i=0, 1 to be the closed set
{ x \in P(a_1,...,a_n) | d(x,p_{a_1,...,a_n,i}) <= eps }
where eps is chosen to guarantee that these two closed
sets are disjoint and have diameter <= 1/(n+1). This shows
that the construction can be carried out for sequences
of length n+1 if it can be done for sequences of length n
and completes the induction. Having constructed all of
these sets, it's then pretty clear what to do next.
Given (a_1, a_2, ... ) in 2^|N, the intersection
P(a_1) \cap P(a_1,a_2) \cap P(a_1,a_2,a_3) ...
is a singleton, say { f(a_1,a_2, ...) }. This
defines (you can check this) a function f: 2^|N --> P
which is clearly one-to-one ...
|> The proof, to me, is very interesting because it uses a different
|> approach than the now old diagonalization argument.
Check carefully, though -- as I noted up above,
it may well _still_ be using the "now old
diagonalization argument" ... After all, in order
to prove that some set is uncountable, you have to
be able to recognize an uncountable set. Most people
(specifically excluding the local cranks) first
acquire that skill by exposure to Cantor's diagonal
argument -- I'm not sure there's any other way to
easily produce a set *known* to be uncountable ...
(I'm sure that last claim is sufficiently provocative
that someone who actually knows what he's talking
about will chime in -- lookin' forward to it ...)
--
Ed Hook | Copula eam, se non posit
Computer Sciences Corporation | acceptera jocularum.
NAS, NASA Ames Research Center | All opinions herein expressed are
Internet: ho...@nas.nasa.gov | mine alone
Doug Norris
>It reminds me of the often asked question:
>Considering that the set of all natural numbers is static
>and infinite, what finite natural number has an infinite
>number of natural numbers less than it?
Often asked by *you*, you mean. Troll.
Doug
Keith Ramsay
ull...@math.okstate.edu (David C. Ullrich) writes:
|>>> I'm not sure about the people who agree with me here either,
|>>> but I suspect there are a lot more people who think of R as an
|>>> unreal abstraction than who think of N that way. Until just now
|>>> I've never heard anyone, well anyone with any credibility at
|>>> all, express any sort of unease regarding induction in N.
|>
|
|we just saw Stephen Montgomery-Smith express unease
|with induction on N - he's _certainly_ a competent
|mathematician, and (although this is less certain) I
|don't _think_ he was just trolling.
Ed Nelson at Princeton has also expressed concerns with the validity
of induction.
There is a standard argument for justifying induction, where we assume
that there exists a class of elements satisfying all the Peano axioms
except for induction. Then we claim induction holds for the subclass
of elements n having the property that every inductive property holds
for n. In his _Predicative Arithmetic_ he disputes this justification
because it's impredicative; it talks about a totality of all properties
of elements while in the process of trying to describe one.
Worries about the real numbers are closely akin to worries about
whether there is such a thing as "all possible properties" or "all
possible subsets".
I think we need to distinguish between attitudes which depart from
actual norms of scientific inquiry, and attitudes which we simply
disagree with, or which we don't consider with the trouble to pursue
in our own research. Science has little to do with learning to accept
anything as absolutely reliable, or even relatively reliable apart
from evidence of reliability.
The way that induction "works" for us is evidence of its having
reliability of *some* kind, but it's difficult to produce stringent
tests of its validity. The logicians have shown that so much of what
we do mathematically can be formulated as deductions in some relatively
"weak" axiom system. How would we go about checking whether when we
see what appear to be mutually confirming facts, this is evidence of
something more than merely the formal consistency of the "weak" axioms
which we've perhaps unwittingly restricted ourselves to using? I think
this topic is worth researching.
For example, I don't know of any better explanation for why certain
theorems of number theory work out in practice, than that we could in
principle count 1, 2, 3, ..., n even for n which are in practice much
too large to count to. We have primality tests for primes p,q which
are much too big for us to count up to them, or even up to their square
roots. Once we've proven (relatively to the usual assumptions) that
they are prime, the machinery of RSA is observed to work. We find that
computing a^{(p-1)(q-1)} modulo pq gives us 1 every time we try it. I
think this serves as confirmation of certain of our mathematical
beliefs-- induction in paricular. But I don't know that anyone has
made a really strong scientific test here. Ideally we should look for
alternative explanations for what we observe, and this seems not to be
the kind of thing mathematicians ordinary try to do.
I think that science works harmoniously with a certain kind of
skeptical attitude, and that we should therefore respect shows of lack
of certainty even in cases where it seems to err too far on the side
of doubt. On the other hand, we shouldn't overvalue attitude as such.
Disbelief in induction, as opposed to mere failure to believe in it
with certainty, seems not to have any evidence on its side.
Keith Ramsay
Jonathan Hoyle
>> and infinite, what finite natural number has an infinite
>> number of natural numbers less than it?
There is no such finite natural number. Hope that answers your
question.
Jonathan Hoyle
Eastman Kodak
Jonathan Hoyle
>> that we adopt more descriptive names for our numbers
We already use names like "imaginary", "irrational", "trascendental", or
just plain "negative" whenever new numbers are discovered. From my
point of view, the number 2 is just as imaginary as the number i is, but
these names have some tradition behind them. Numbers are conceptual
(take two apples, remove the apples, where is the two-ness?).
However, if you really need more rigorous definitions for numbers, set
theorists will be happy to give it to you in terms of sets. Thus, you
no longer struggle with the concept of number, only with the concept of
set.
david_...@my-deja.com
kra...@aol.commangled (Keith Ramsay) wrote:
> In article <39649483....@nntp.sprynet.com>,
> ull...@math.okstate.edu (David C. Ullrich) writes:
> |>>> I'm not sure about the people who agree with me here either,
> |>>> but I suspect there are a lot more people who think of R as an
> |>>> unreal abstraction than who think of N that way. Until just now
> |>>> I've never heard anyone, well anyone with any credibility at
> |>>> all, express any sort of unease regarding induction in N.
> |>
> |>This might be a good characterization of credibility. :-)
> |
> | I would have said the same a few days ago. But
> |we just saw Stephen Montgomery-Smith express unease
> |with induction on N - he's _certainly_ a competent
> |mathematician, and (although this is less certain) I
> |don't _think_ he was just trolling.
>
> Ed Nelson at Princeton has also expressed concerns with the validity
> of induction.
So I'd heard. But I'd never heard much about him other than
that he had curious attitudes about the integers - otoh with
M-S I have independent reasons for regarding him as a
competent mathematician. (This is not to say that Nelson
isn't, just a statement about what I happen to know from
my own experience.)
> There is a standard argument for justifying induction, where we assume
> that there exists a class of elements satisfying all the Peano axioms
> except for induction. Then we claim induction holds for the subclass
> of elements n having the property that every inductive property holds
> for n. In his _Predicative Arithmetic_ he disputes this justification
> because it's impredicative; it talks about a totality of all
properties
> of elements while in the process of trying to describe one.
Yeah, that would be a problem. Good thing it's obviuous
that induction works...
Well _I_ wasn't the guy who suggested that belief in
induction could be a prerequisite for credibility - I may have
admitted to once having had such a view but I disclaimed it
before ever uttering it.
But I do find the validity of induction, not in some
formal system, but for statements about the "real" integers,
somewhat un-disbelievable. This is not the sort of thing I
would attempt to support rigorously, because (for example)
I would not want to attempt to define what the _real_
integers are. (It seems possible I'm beginning to see
Kronecker's point: Pertti created the integers, all else
is the work of mud peoples.)
> Keith Ramsay
Lee Rudolph
> kra...@aol.commangled (Keith Ramsay) wrote:
>>
>> Ed Nelson at Princeton has also expressed concerns with the validity
>> of induction.
>
> So I'd heard. But I'd never heard much about him other than
>that he had curious attitudes about the integers - otoh with
>M-S I have independent reasons for regarding him as a
>competent mathematician. (This is not to say that Nelson
>isn't, just a statement about what I happen to know from
>my own experience.)
Years before he (publically; or at least, in print--I have no
idea what he might have said in the Fine Hall common room)
"expressed concerns with the validity of induction", he was
a notable analyst of a somewhat algebraic bent. I have several
of his books from the old yellow Princeton "Mathematical Notes"
series, in particular _Tensor Analysis_, in which he works out
the example of the 4-dimensional configuration space M of a
(rather idealized) automobile. "There are two distinguished
vector fields, called Steer and Drive, on M corresponding
to the two ways in which we can change the configuraiton of
a"nd two more which he calls Slide and Rotate. After some
calculations, "the Lie product of Steer and Drive is equal
to Slide + Rotate on" the submanifold of M where the wheels
are pointing in the same direction as the car. "Let
us denote the Lie product of Steer and Drive by Wriggle.
Then further simple computations show that we have the
commutation relations
[Steer, Drive] = Wriggle,
[Steer, Wriggle] = -Drive
[Wriggle, Drive] = Slide,
and the commutator of Slide with Stter, Drive, and Wriggle is
zero. Thus the four vector fields span a four dimensional
solvalble lie algebra over R. To get out of an extremely
tight parking spot, Wriggle is insufficient because it may
produce too much rotation. The last commutation relation shows,
howeer, that one may get out of an arbitrarily tight
parking spot in the following way: wriggle, drive, reverse
wriggle (this requires a cool head), reverse drive, wriggle,
drive, ... ."
All in all a wonderful illustration of holonomy.
Lee Rudolph
Martin Vaeth
>> It is the statement that any subset of a complete separable metric space has
>> the property of Baire
Thanks for the proof. I never dealt much with AD, because I do not feel so
convenient about the consistency of large cardinals.
>> [...] if it turns out that the only
>> mathematical objects relevant for physics are the constructible ones,
>> it would probably be more *convenient* to just throw AC overboard
>> (actually, this is what I mean when I say that AC does not hold).
>
>Not at all, mon vieux -- this is the whole point! Considerations of
>the wider universe V give us insight into the behavior of L(R) which
>would be difficult otherwise to come by.
Yes and no. I do not say it is useless to ever study powerful axioms like
AC/large cardinals/... and their consequences. However, IMHO the results you
get in this way are more of "academic" interest in the sense that they show
the *limits* of L(R). It appears to me that for statements of physical interest
in L(R) you really have to confine to proofs without AC.
But maybe I just do not know enough examples of such statements...
(set theory is only a hobby I ran across, not my field of research).
>For example, the large cardinals in V give us that L(R) satisfies AD.
This is an example for such an "academic" statement: It is "useless" for
physics, because we do not know about the truth (or even consistency) of
large cardinals. (Nevertheless, it is of course very interesting).
BTW: Do you mind posting a reference for this result? As I understood,
this is what Solovay had conjectured, but he had "only" been able to prove that
large cardinals imply weaker statement (like measurability of all subsets of
the reals etc).
>Because L(R) sits inside V in such a nice, stable way [...]
>there's very little cost associated with finding out about
>L(R) by reasoning with V but restricting quantifiers when necessary,
>rather than refusing ever to consider sets outside L(R).
I doubt that if you have, say, a complicated numerical algorithm whose
convergence proof uses the existence of a certain separating functional
which you obtain by Hahn-Banach in l_\infty (and you just can not *find*
another proof) that you are really able to judge whether the convergence
happens in L(R). (Replace "convergence proof of numerical algorithm" by
"existence proof for an equation" if you prefer).
Clark
david_...@my-deja.com wrote:
>
> In article <20000710013421...@nso-cv.aol.com>,
> kra...@aol.commangled (Keith Ramsay) wrote:
> > In article <39649483....@nntp.sprynet.com>,
> > ull...@math.okstate.edu (David C. Ullrich) writes:
...
> > |>>> Until just now
> > |>>> I've never heard anyone, well anyone with any credibility at
> > |>>> all, express any sort of unease regarding induction in N.
> > |>
> > |>This might be a good characterization of credibility. :-)
> > |
> > | I would have said the same a few days ago. But
> > |we just saw Stephen Montgomery-Smith express unease
> > |with induction on N - he's _certainly_ a competent
> > |mathematician, and (although this is less certain) I
> > |don't _think_ he was just trolling.
> >
> > Ed Nelson at Princeton has also expressed concerns with the validity
> > of induction.
>
> So I'd heard. But I'd never heard much about him other than
> that he had curious attitudes about the integers - otoh with
> M-S I have independent reasons for regarding him as a
> competent mathematician. (This is not to say that Nelson
> isn't, just a statement about what I happen to know from
> my own experience.)
>
> > There is a standard argument for justifying induction, where we assume
> > that there exists a class of elements satisfying all the Peano axioms
> > except for induction. Then we claim induction holds for the subclass
> > of elements n having the property that every inductive property holds
> > for n. In his _Predicative Arithmetic_ he disputes this justification
> > because it's impredicative; it talks about a totality of all
> properties
> > of elements while in the process of trying to describe one.
Coming at this from a slightly different angle, there's a (?) well-known
justification (2nd order) of Peano's axioms (including induction) by
Frege from Hume's Principle (number of F's = number of G's iff F's can
be one-one correlated with G's). (Hume's Principle doesn't suffer the
fate of Frege's axiom V, as various people - Boolos, Burgess, Hodes ...
- have noted.)
Question: is Hume's Principle more, or less, believable than the
principle of induction? (I'd like to know what people think ...)
Bob
Stephen Montgomery-Smith
>
> ...................
>
> I think we need to distinguish between attitudes which depart from
> actual norms of scientific inquiry, and attitudes which we simply
> disagree with, or which we don't consider with the trouble to pursue
> in our own research. Science has little to do with learning to accept
> anything as absolutely reliable, or even relatively reliable apart
> from evidence of reliability.
>
>
> I think that science works harmoniously with a certain kind of
> skeptical attitude, and that we should therefore respect shows of lack
> of certainty even in cases where it seems to err too far on the side
> of doubt. On the other hand, we shouldn't overvalue attitude as such.
> Disbelief in induction, as opposed to mere failure to believe in it
> with certainty, seems not to have any evidence on its side.
>
Thanks. I was getting a little disturbed at being labeled a crackpot.
Stephen Montgomery-Smith
>
> Coming at this from a slightly different angle, there's a (?) well-known
> justification (2nd order) of Peano's axioms (including induction) by
> Frege from Hume's Principle (number of F's = number of G's iff F's can
> be one-one correlated with G's). (Hume's Principle doesn't suffer the
> fate of Frege's axiom V, as various people - Boolos, Burgess, Hodes ...
> - have noted.)
I would be interested to know this argument. If it is short, could
you share it on the internet?
Stephen
ca314159
kra...@aol.commangled (Keith Ramsay) wrote:
> There is a standard argument for justifying induction, where we assume
> that there exists a class of elements satisfying all the Peano axioms
> except for induction. Then we claim induction holds for the subclass
> of elements n having the property that every inductive property holds
> for n. In his _Predicative Arithmetic_ he disputes this justification
> because it's impredicative; it talks about a totality of all
properties
> of elements while in the process of trying to describe one.
The is a justification for watching your back when it comes to
logic. Especially when you try to map it onto 'realty' with models.
In a previous episode, our hero Erwin Schrodinger, was caught
in a death defying game of Quantum Roulette:
Tick, tick, tick. Oiiink!
"Ooooohh, sorry Elwood, you landed in-between
'Mind-Body Problem', I'm afraid that sets you
back over two thousand years..."
"My name's Erwin."
"...If the wheel had landed on Bluetooth instead,
you would have won a weekend-for-two at the
hoe-down with your cousin Fanny Mae. But we do
have a consolation prize, you get this fine
wireless network made from Teligent-Remec
microwave repeaters and a new ballcock for
your Ford Woody so you can tow that trailer of
yours to a nice spot down the road. How about
that Elwood ?!."
"My name's Erwin."
Elementar- und Volkergedanken
Goethe's Faust, Adolf Bastian, Memetics:
http://www.uni-ulm.de/uni/intgruppen/memosys/desn27.htm
"Go", an addictive game:
http://www.cwi.nl/people/jansteen/go/go.html
(so is the threaded version of Go: Usenet, which doesn't
have a "Go Back" button, neither do birth canals nor
detonator buttons)
"What do the Spaces mean?":
http://www.spe.sony.com/movies/Jumanji/contest.html
Glass gives light, shelter, and food for thought.
"Don't walk on the Glass !"
- A silicon warning sign
- Terry Hinely and Glasnots
You've made it this far, you may pass "Go" and collect $200.
- OF ZERO NOTHINGS -----------------------------------------
In applied vector mathematics there are at least three basic kinds
of "zero":
1) ---->0<----
2) <----0---->
3) 0
each of which have a zero "resultant".
A voltage may be zero if +100V is added to -100V as in 1),
or if -100V is added to +100V as in 2),
or if both of the two voltages are zero themselves, as in 3).
But, 3) is not the same as saying "nothing", 3) may
be the zero resultant of zero forces which themselves may be
the resultants of non-zero forces. Therefore, an additional
vector category is needed
4) nothing, void, nada, nih...
to represent a mythical "nothingness" which cannot be
measured, in the same sense that infinity cannot be
measured.
1) 2) and 3) are all measureable. 4) is not measureable
without infinite knowledge, something we expect of a god,
and hence knowledge of "nothing" is considered a
quasi-godlike state of mind in the East.
Black is the absense of color. White is the lack of color.
The "color" black is silent, the "color" white is noisy.
Gray. Is the cup half-empty, or is it half-full ?
"We are the Knights of Nih" (MIB)
- Monty Python and the Holy Grail
http://www.intriguing.com/mp/pictures/grail/shrubber.jpg
----------------------------------------------------------
anfscu:
Everyone has a supercomputer inside and outside their
Klein bottle shaped head:
http://www.ancientsites.com/~Mirjam_Nebet/coron3.html
http://www.infidels.org/library/historical/thomas_paine/origin_free-maso
nry.html
"Watch out for those Xerox men !
I've seen them, those X-Men.
They look like bad copies,
worse each time through.
They waft of foul chemicals.
Quantum Men in B&W.
They'll copy you !"
- Chief Spoon Tongue
http://www.columbia.edu/cu/libraries/indiv/dsc/intell.html
"My big brother will beat up your Big Brother."
- Anonymous
--
http://www.bestweb.net/~ca314159/
There's two kinds of work: working and networking.
One is to get paid now, the other is to get paid tomorrow.
Clark
Stephen Montgomery-Smith wrote:
>
> Clark wrote:
> >
> > Coming at this from a slightly different angle, there's a (?) well-known
> > justification (2nd order) of Peano's axioms (including induction) by
> > Frege from Hume's Principle (number of F's = number of G's iff F's can
> > be one-one correlated with G's). (Hume's Principle doesn't suffer the
> > fate of Frege's axiom V, as various people - Boolos, Burgess, Hodes ...
> > - have noted.)
>
> I would be interested to know this argument. If it is short, could
> you share it on the internet?
The essence of it is short enough, I think. (I'll give references for
details below.)
Hume's Principle:
Nx:Fx=Nx:Gx iff (ER)(Fx1-1(R)Gx)
['N'say'number of' ...'1-1(R)'say'1-1 related by R to' (standardly
defined)]
First, use Hume's Principle to get cardinality by abstraction:
(F)(Ey)(y=Nx:Fx)
Next, define
zero [0=Nx:x!=x] and
x Precedes y [Pxy iff (EF)(Ez)(Fz&y=Nw:Fw&x=Nv:(Fv&v!=z))]
[Prove that P is 1-1]
Now define the ancestral relation R* of a relation R:
R*xy iff (F)[((z)(Rxz->Fz)&(v)(w)((Fv&Rvw)->Fw))->Fy]
[Prove that Rxy->R*xy and that R* is transitive]
So we can define natural number:
Natx iff (0=x or P*0x)
And now it's easy to see that induction
(F)[F(0)&(x)(Fx->(y)(Pxy->F(y)))->(x)(Natx->Fx)]
follows
[take each conjunct of the antecedent of P*0x separately ... deduce Fx]
This might seem a bit of a damp squib at first. Hasn't Frege just
defined natural numbers as things that induction works on by means of
the more general notion of ancestral? There may be something in this.
But, on the other hand, doing the work this way gives us a view of the
Peano axioms as a whole (the other axioms can also be derived as logical
consequences of Natx), and as consequences just of the notion of
cardinals with ancestral precedence.
Now it looks as though the only non-logical notion in play here is just
that of Hume's Principle. Is it true ... if so, is it more or less
obvious than the principle of induction?
Frege, of course, tried - and, as Russell showed him, failed - to derive
Hume's Principle logically.
[References: Much of the above can be found at slightly greater length
in Crispin Wright's 'Frege's Conception of Numbers as Objects', (chapter
4) which pretty much set the whole neo-Fregean ball rolling, I think.
Many relevant subsequent papers are collected in William Demopoulos
(ed.) 'Frege's Philosophy of Mathematics'.]
I'm much more of a neophyte than an expert with all this, by the way -
and I'd appreciate criticism by those who are more expert, of which I'm
sure there are many around.
Bob
ca314159
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
> Keith Ramsay wrote:
> >
> > ...................
> >
> > I think we need to distinguish between attitudes which depart from
> > actual norms of scientific inquiry, and attitudes which we simply
> > disagree with, or which we don't consider with the trouble to pursue
> > in our own research. Science has little to do with learning to
accept
> > anything as absolutely reliable, or even relatively reliable apart
> > from evidence of reliability.
> >
> > ....................
> >
> > I think that science works harmoniously with a certain kind of
> > skeptical attitude, and that we should therefore respect shows of
lack
> > of certainty even in cases where it seems to err too far on the side
> > of doubt. On the other hand, we shouldn't overvalue attitude as
such.
> > Disbelief in induction, as opposed to mere failure to believe in it
> > with certainty, seems not to have any evidence on its side.
> >
>
> Thanks. I was getting a little disturbed at being labeled a crackpot.
If it completely matters to someone, what other people think,
then that someone is usually labelled a "crack-pot". This is
the "Truth is not a democracy. Truth is universal." perspective.
If it doesn't matter to someone at all what others think,
then that someone is also, usually labelled a "crack-pot".
This is the "Nothing is absolute. Everything is relative."
perspective.
Which crackpot flavor were you thinking of ?
Does it have syrup or nuts on it.
Anonymous
<the_grea...@my-deja.com> wrote:
>That reminds me of a comment Cantor made while
>being held captive at an insane asylum in germany:
>
>"Just because 0+0 equals zero, doesn't mean that
>0+0+0+... equals zero. In fact, I can prove
>0+0+0+... equals whatever I want it to."
>
>So, for anyone who's actually trying to understand
>these things, consider: Cantorians _still_ believe that!
Nathan, stop making things up. Read Asop's fable:
The_Boy_Who_Cried_Wolf.
As usual, you're facts are all wrong. Cantor was never in
an insane asylum. He once visitting a mental institution,
but only in the capacity of consulting doctor. BTW, his
stay was brief, it only lasted while he was on vacation and
the hospital needed his services.
Why do you keep misquoting Cantor? He never said, "for
sufficiently large 0's, 0+0 = oo" Of course _that_ would be
preposterous. Instead Cantor made the logical observation
that, "for sufficiently _many_ zeros, 0+0+0+... = oo"
Insignificant I
--------== Posted Anonymously via Newsfeeds.Com ==-------
Featuring the worlds only Anonymous Usenet Server
-----------== http://www.newsfeeds.com ==----------
Nathan the Great
Anonymous <nob...@newsfeeds.com> wrote:
> As usual, you're facts are all wrong. Cantor was never in
> an insane asylum. He once visitting a mental institution,
> but only in the capacity of consulting doctor. BTW, his
> stay was brief, it only lasted while he was on vacation and
> the hospital needed his services.
According to historians, Cantor was the town drunk,
not a doctor. If you want me to believe Cantor was a
doctor, you need to supply additional information.
Doug Norris
>In article <396ccfbb...@anonymous.usenet.com>,
> Anonymous <nob...@newsfeeds.com> wrote:
>> As usual, you're facts are all wrong. Cantor was never in
>> an insane asylum. He once visitting a mental institution,
>> but only in the capacity of consulting doctor. BTW, his
>> stay was brief, it only lasted while he was on vacation and
>> the hospital needed his services.
>According to historians, Cantor was the town drunk,
>not a doctor. If you want me to believe Cantor was a
>doctor, you need to supply additional information.
Talking to yourself again, Nate?
Doug
Richard Carr
:Date: Fri, 14 Jul 2000 07:50:36 GMT
:From: Nathan the Great <the_grea...@my-deja.com>
:Newsgroups: sci.math
:Subject: Re: Axiom of Choice
:
:In article <396ccfbb...@anonymous.usenet.com>,
: Anonymous <nob...@newsfeeds.com> wrote:
:> As usual, you're facts are all wrong. Cantor was never in
:> an insane asylum. He once visitting a mental institution,
:> but only in the capacity of consulting doctor. BTW, his
:> stay was brief, it only lasted while he was on vacation and
:> the hospital needed his services.
:
:According to historians, Cantor was the town drunk,
:not a doctor. If you want me to believe Cantor was a
:doctor, you need to supply additional information.
:
:Nathan the Great
:Age 12
:
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Nathan the Great
a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
Dave, you don't fool me. Behind those mild
manners lurks a diabolical fiend, sitting
motionless at the center of every web of
confusion spun by the evil Cantorians.
> Each transposition is a transformation that preserves
> bijections. Without much trouble, you can show that
> the composition of any two such transpositions is a
> transformation that preserves bijections. Likewise
> for the composition of any three, and so on.
>
> Exactly how for does that "and so on" apply?
It is tempting to say the "and so on" means
that the process of proceeding to the successor
may be repeated any finite number of times; but
"finite number" lacks a precise definition,
therefore it is best not to use this notion.
- Bertrand Russell
> In other words, if we let P(n) represent the
> statement "every composition of n transpositions
> yields a transformation that preserves bijections,
> then we can ask for which values of n is P(n) true.
> With a little thought, we can show:
>
> 1. P(1).
> 2. P(n) -> P(n+1) for all n.
>
> Using mathematical induction, we draw the conclusion:
> P(n) holds for all __________ n. I have left a blank
> in that sentence for you to fill in.
My answer: INDUCTIVE.
The phrase "inductive numbers" means the same as
the hitherto spoken of "natural numbers." But using
the phrase "inductive numbers" is preferable, as it
reminds us that those Sets are defined via mathematical
induction.
The use of mathematical induction in demonstrations
was, in the past, something of a mystery. There
seemed no reasonable doubt that it was a valid method
of proof, but no one quite knew why it was valid. We
now know that mathematical induction is a definition,
not a principle. There are some numbers to which it
can be applied, and there are others to which it
cannot. We define the "natural numbers" as those to
which proofs by mathematical induction can be applied,
i.e. as those that possess all inductive properties.
It follows that such proofs can be applied to the
natural numbers, not in virtue of any mysterious
intuition or axiom or principle, but as a purely
verbal proposition. If "quadrupeds" are defined as
animals having four legs, it will follow that animals
that have four legs are quadrupeds; and the case of
numbers that obey mathematical induction is exactly
similar.
> This is a test of your understanding of mathematical
> induction.
Dave, you failed your own test. There are two
types of induction. My proof uses "complete
induction", not "simple induction" (the kind
you demonstrated above).
I'd like to see things from your point of view,
but my head won't shrink to that size.
Simple Induction is a common method of proving that
each of an infinite sequence of mathematical statements
is true by proving that (1) the first statement is
true; (2) the truth of any one of the statements
always implies the truth of the next one.
Complete Induction is less concerned with proving
individual statements than it is with proving
that those statements form a _complete_ inductive
Set. To use complete induction it is necessary
to show that:
(1) the first statement is true. (In this case,
the first statement is: "an unspecified
bijection between N and FP(N) exists" BTW,
the truth of that statement is the starting
assumption)
(2) for each n the truth of every statement, from
the first to the nth inclusive, would imply the
truth of the (n+1)th. (In this case, the truth
of a valid partially specific bijection (one
which pairs the first n elements of N to elements
of FP(N), using my specific match-up scheme)
implies the truth of a larger specific bijection,
one which also includes the (n+1)th element of N)
Here is what I wrote:
>>To convert the original bijection into the one
>>above, a simple step-by-step process, guaranteed
>>not to corrupt the bijection status, is used:
>>
>>Starting with 1 and counting 1,2,3, etc. find the
>>first element of N that is not correctly (as shown
>>above) paired to a set in FP(N). Call this the nth
>>element of N. It should be paired to an n-element
>>set in FP(N), having 1 as its 1st element and n as
>>its n-th element. Once the correct set is located
>>(it has to be somewhere in FP(N)), swap the
>>incorrectly paired set, with the correct one.
>>Continue this process for all members of N.
>>
>>Now, every member of N is paired, as shown above.
Dave, since you do not believe the process
can be extrapolated, to include _all_ members
of N, which members of N, are not validly
paired, when the pairing scheme finishes?
According to the Well Ordering Principle,
this Set must either contain a least member
or be empty.
The possibilities:
(1) Set is empty.
(2) Set has a least member.
(3) No such Set. (the process never ends)
My replies:
(1) Contradicts your present argument and
reaffirms the inconsistency (countable =
uncountable) we discussed in the past.
(2) This is not possible. In the proposed final
bijection, every n in N is paired with a
finite subset {1,2,3...n} of FP(N). If you
insist that the proposed pairings can't be
formed for all n, valid reasons for that
belief should be given. Here are two invalid
reasons: (1) for some n, the Set {1,2,3...n}
does not exist and (2) for some n, the Set
{1,2,3...n} is already _correctly_ paired to
a member of N that is less than n.
(3) That 'process' was just my way of explaining
complete induction. My intention wasn't to
portray the process as an infinite repetition
of actual manipulations, but instead, as the
two steps of a complete induction thought
experiment. So, when you say, "the process
never ends", you mean, "complete induction is
a sham."
Note: Complete induction can be rephrased as:
if a Set contains 1 and, for each n, it
contains n+1 whenever it contains all
numbers less than n+1, then it must contain
every natural number.
Nathan the Great
Age 12
--------------------------------------------------
Weak arguments are often thrust before my path; but
although they are most unsubstantial, it is not easy
to destroy them. There is not a more difficult feat
known than to cut through a cushion with a sword.
- Richard Whately
--
Nathan the Great
Age 12
Dave Seaman
>Dave, you don't fool me. Behind those mild
>manners lurks a diabolical fiend, sitting
>motionless at the center of every web of
>confusion spun by the evil Cantorians.
>> Each transposition is a transformation that preserves
>> bijections. Without much trouble, you can show that
>> the composition of any two such transpositions is a
>> transformation that preserves bijections. Likewise
>> for the composition of any three, and so on.
>> Exactly how for does that "and so on" apply?
>It is tempting to say the "and so on" means
>that the process of proceeding to the successor
>may be repeated any finite number of times; but
>"finite number" lacks a precise definition,
>therefore it is best not to use this notion.
> - Bertrand Russell
For our purposes, the "finite numbers" are the natural numbers, which are
the members of the smallest set that contains 0 and is closed under the
successor operation s(n) = n U {n}.
>> In other words, if we let P(n) represent the
>> statement "every composition of n transpositions
>> yields a transformation that preserves bijections,
>> then we can ask for which values of n is P(n) true.
>> With a little thought, we can show:
>> 1. P(1).
>> 2. P(n) -> P(n+1) for all n.
Actually I can say P(0) is true as a starting point, because a
composition of 0 transpositions yields the identity transformation, which
is a bijection.
>> Using mathematical induction, we draw the conclusion:
>> P(n) holds for all __________ n. I have left a blank
>> in that sentence for you to fill in.
>My answer: INDUCTIVE.
I had in mind "positive integers" or possibly "natural numbers", but your
answer works to the extent that we can say P holds for all n in a set
that contains 0 (or 1) and is closed under the successor operation.
Whatever name you use, it seems we agree that P(n) has not been
demonstrated to hold (not by the induction argument, anyway) for n = w.
In fact, your own argument provides a counterexample showing that P(w) is
false.
> The phrase "inductive numbers" means the same as
>the hitherto spoken of "natural numbers." But using
>the phrase "inductive numbers" is preferable, as it
>reminds us that those Sets are defined via mathematical
>induction.
>> This is a test of your understanding of mathematical
>> induction.
>Dave, you failed your own test. There are two
>types of induction. My proof uses "complete
>induction", not "simple induction" (the kind
>you demonstrated above).
There are two kinds of induction, all right. One is simple induction,
which we have been discussing, and the other is transfinite induction.
>Simple Induction is a common method of proving that
>each of an infinite sequence of mathematical statements
>is true by proving that (1) the first statement is
>true; (2) the truth of any one of the statements
>always implies the truth of the next one.
>Complete Induction is less concerned with proving
>individual statements than it is with proving
>that those statements form a _complete_ inductive
>Set. To use complete induction it is necessary
>to show that:
>(1) the first statement is true. (In this case,
> the first statement is: "an unspecified
> bijection between N and FP(N) exists" BTW,
> the truth of that statement is the starting
> assumption)
>(2) for each n the truth of every statement, from
> the first to the nth inclusive, would imply the
> truth of the (n+1)th. (In this case, the truth
> of a valid partially specific bijection (one
> which pairs the first n elements of N to elements
> of FP(N), using my specific match-up scheme)
> implies the truth of a larger specific bijection,
> one which also includes the (n+1)th element of N)
That's just a variation on simple induction. It still applies only to
the natural numbers and not to transfinite ordinals.
In order to get transfinite induction, you need to show that for every
ordinal alpha, (P(beta) for all beta < alpha) -> P(alpha). For your
proposition P this fails in the case alpha = w, as your own argument
demonstrates.
>Here is what I wrote:
>>>To convert the original bijection into the one
>>>above, a simple step-by-step process, guaranteed
>>>not to corrupt the bijection status, is used:
>>>Starting with 1 and counting 1,2,3, etc. find the
>>>first element of N that is not correctly (as shown
>>>above) paired to a set in FP(N). Call this the nth
>>>element of N. It should be paired to an n-element
>>>set in FP(N), having 1 as its 1st element and n as
>>>its n-th element. Once the correct set is located
>>>(it has to be somewhere in FP(N)), swap the
>>>incorrectly paired set, with the correct one.
>>>Continue this process for all members of N.
>>>Now, every member of N is paired, as shown above.
>Dave, since you do not believe the process
>can be extrapolated, to include _all_ members
>of N, which members of N, are not validly
>paired, when the pairing scheme finishes?
>According to the Well Ordering Principle,
>this Set must either contain a least member
>or be empty.
Each member of N is swapped only finitely many times, but some members of
FP(N) are swapped infinitely many times. This means some members of
FP(N) that were originally in the list are left unpaired after all the
transpositions have taken place. The bijection is not preserved.
Nathan the Great
a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
>>> In other words, if we let P(n) represent the
>>> statement "every composition of n transpositions
>>> yields a transformation that preserves bijections,
>>> then we can ask for which values of n is P(n) true.
>>> With a little thought, we can show:
>>>
>>> 1. P(1).
>>> 2. P(n) -> P(n+1) for all n.
>>>
>>> P(n) holds for all __________ n. I have left a blank
>>> in that sentence for you to fill in.
>>
>>My answer: INDUCTIVE.
>
> not been demonstrated to hold (not by the induction
> argument, anyway) for n = w.
Dave, I wish you would address my comments instead
of criticizing your own. There is no "P(n)" in
what I wrote. Our arguments use different types
of induction. (simple vs. complete)
Also, stop saying 'we agree' unless we actually do.
True or not, your one-sided agreement is dishonest.
Here is something you snipped:
"Complete induction can be rephrased as:
if a Set contains 1 and, for each n, it
contains n+1 whenever it contains all
numbers less than n+1, then it must
contain every natural number."
Why was that silently removed?
If there are legitimate reasons to dispute
that quotation, please send them to:
Or _PENGUIN_DICTIONARY_OF_MATHEMATICS_ CO:
Derek Gjertsen University of Leverpool
Nick Higham University of Manchester
Terence Jackson University of York
Mark Pollicott University of Manchester
Elmer Rees University of Edinburgh
Peter Sprent University of Dendee
David Nelson University of Cambridge
> In fact, your own argument provides a
> counterexample showing that P(w) is false.
Bah, there is no P(w) in my proof.
Besides, even if I do contradict myself with
a counterexample, that still doesn't justify
your claim that my argument is invalid. My
argument could still be sound. The axioms
might, in fact, be causing the inconsistency.
After all this time, I thought at least you,
Dave, would understand that I *must* contradict
my own argument, with a valid counterexample,
to achieve my goal (inconsistency proof).
Pertti, this is your turf, speak up.
Dave Seaman
> a...@seaman.cc.purdue.edu (Dave Seaman) wrote:
>>>> In other words, if we let P(n) represent the
>>>> statement "every composition of n transpositions
>>>> yields a transformation that preserves bijections,
>>>> then we can ask for which values of n is P(n) true.
>>>> With a little thought, we can show:
>>>> 1. P(1).
>>>> 2. P(n) -> P(n+1) for all n.
>>>> Using mathematical induction, we draw the conclusion:
>>>> P(n) holds for all __________ n. I have left a blank
>>>> in that sentence for you to fill in.
>>>My answer: INDUCTIVE.
The answer I had in mind was "finite". However, I will accept your
answer with the understanding that an inductive set is one that satisfies
PA, particularly the fifth axiom that says there are no numbers other
than 1 and its successors.
The point I was trying to make is that the induction scheme presented
here does not extend to infinite values of n. That requires transfinite
induction, which is another matter entirely.
>> Whatever name you use, it seems we agree that P(n) has
>> not been demonstrated to hold (not by the induction
>> argument, anyway) for n = w.
That is, w is not finite and ordinary induction does not extend to
infinite values.
>Dave, I wish you would address my comments instead
>of criticizing your own. There is no "P(n)" in
>what I wrote. Our arguments use different types
>of induction. (simple vs. complete)
I explained what P(n) means in the passage that you quoted. This is not
about power sets, by the way. You were talking about compositions of
transpositions, and you made the implicit claim that infinite
compositions are well-defined.
Your "simple vs. complete" is merely a characterization of two different
ways of formulating ordinary induction. Either way, it does not extend
to infinite values.
>Also, stop saying 'we agree' unless we actually do.
>True or not, your one-sided agreement is dishonest.
>Here is something you snipped:
> "Complete induction can be rephrased as:
> if a Set contains 1 and, for each n, it
> contains n+1 whenever it contains all
> numbers less than n+1, then it must
> contain every natural number."
I rest my case. You just admitted that your form of induction applies
only to natural numbers, and therefore it does not extend to omega.
>Why was that silently removed?
Because ordinary induction is ordinary induction, and you have not
established P(w) by your argument.
>If there are legitimate reasons to dispute
>that quotation, please send them to:
Why would I dispute the very quotation in which you admitted the point
that I was trying to establish?
>> In fact, your own argument provides a
>> counterexample showing that P(w) is false.
>Bah, there is no P(w) in my proof.
You claimed that transpositions can be infinitely composed. That is
P(w).
>Besides, even if I do contradict myself with
>a counterexample, that still doesn't justify
>your claim that my argument is invalid.
Your argument is invalid because you claimed to have proved P(w) with
ordinary (not transfinite) induction.