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.