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| Aatu Koskensilta... |
 Posted: Mon Oct 12, 2009 12:16 pm |
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Guest
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It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
--
Aatu Koskensilta (aatu.koskensilta at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Nam Nguyen... |
| Posted: Mon Oct 12, 2009 12:29 pm |
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Aatu Koskensilta wrote:
[quote:9979401a0b]It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
[/quote:9979401a0b]
Don't have a every details yet but it's possible the order be
relativistic over the (more absolute) existences of the reals,
which could give rise to paradoxes if we insist there's a "global"
absolute well order. Imho. |
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| Aatu Koskensilta... |
| Posted: Mon Oct 12, 2009 12:29 pm |
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Nam Nguyen <namducnguyen at (no spam) shaw.ca> writes:
[quote:390abe550e]Don't have a every details yet but it's possible the order be
relativistic over the (more absolute) existences of the reals, which
could give rise to paradoxes if we insist there's a "global" absolute
well order. Imho.
[/quote:390abe550e]
Whatever the merits of this, it appears to be a theoretical
(philosophical?) doctrine or hypothesis, not an intuition contradicted
by the well-ordering theorem.
--
Aatu Koskensilta (aatu.koskensilta at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Nam Nguyen... |
| Posted: Mon Oct 12, 2009 1:46 pm |
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Aatu Koskensilta wrote:
[quote:405241af82]Nam Nguyen <namducnguyen at (no spam) shaw.ca> writes:
Don't have a every details yet but it's possible the order be
relativistic over the (more absolute) existences of the reals, which
could give rise to paradoxes if we insist there's a "global" absolute
well order. Imho.
Whatever the merits of this, it appears to be a theoretical
(philosophical?) doctrine or hypothesis, not an intuition contradicted
by the well-ordering theorem.
[/quote:405241af82]
It's only as much as philosophical as the current mathematical reasoning
based on the assumed knowledge of the naturals is. If you already
preconceived the truth of well-order of the reals isn't philosophically
based and isn' counterintuitive, well then there would be - as you said -
"mysteries of sets" you'd miss.
If you keep an open-minded "attitude", I'd further elaborate. |
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| Rupert... |
| Posted: Mon Oct 12, 2009 2:21 pm |
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On Oct 13, 5:16 am, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote:073d30b63d]It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
[/quote:073d30b63d]
Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive. |
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| Aatu Koskensilta... |
| Posted: Mon Oct 12, 2009 6:24 pm |
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Rupert <rupertmccallum at (no spam) yahoo.com> writes:
[quote:10708ae407]Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive.
[/quote:10708ae407]
Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense. That aside, I was wondering specifically about the claim that the
well-ordering theorem itself is counter-intuitive, as alluded to in
e.g. the famous quip
The axiom of choice is obviously true, the well-ordering theorem
obviously false -- and who can tell of Zorn's lemma?
Just what intuitions are contradicted by the well-ordering theorem?
--
Aatu Koskensilta (aatu.koskensilta at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Nam Nguyen... |
| Posted: Mon Oct 12, 2009 10:21 pm |
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Aatu Koskensilta wrote:
[quote:c63023af0e]Rupert <rupertmccallum at (no spam) yahoo.com> writes:
Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive.
Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense. That aside, I was wondering specifically about the claim that the
well-ordering theorem itself is counter-intuitive, as alluded to in
e.g. the famous quip
The axiom of choice is obviously true, the well-ordering theorem
obviously false -- and who can tell of Zorn's lemma?
Just what intuitions are contradicted by the well-ordering theorem?
[/quote:c63023af0e]
It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi),
where M(x) is the Major number of x and Pi and e are the 2 well known
transcendentals (please refer to the thread "Ttranscendental Goldbach
Conjecture"), since nobody would have a slightest intuition which one
assertion is the case.
Since the well-ordering of reals implies one assertion over the other
is true, it's therefore counterintuitive.
Put it differently, it's quite possible it's impossible to know which
inequality would hold - in all models of reals. It's therefore counter-
intuitive to talk about the well-ordering of the whole set of reals. |
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| Herman Jurjus... |
| Posted: Tue Oct 13, 2009 3:32 am |
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Aatu Koskensilta wrote:
[quote:9d01a4028f]Rupert <rupertmccallum at (no spam) yahoo.com> writes:
Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive.
Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense. That aside, I was wondering specifically about the claim that the
well-ordering theorem itself is counter-intuitive, as alluded to in
e.g. the famous quip
The axiom of choice is obviously true, the well-ordering theorem
obviously false -- and who can tell of Zorn's lemma?
Just what intuitions are contradicted by the well-ordering theorem?
[/quote:9d01a4028f]
Imho, the quip tries to express something completely different.
It's not an expression of mathematician's intuitions about sets.
Mathematicians don't care about sets - they care about mathematics.
And a general purpose foundation for mathematics should ideally not turn
something into an indispensable truth that is, for mathematics, quite
dispensable.
Btw, personally i find Freyling's argument quite appealing. But i also
think that AD and AC are both true, so you should better not take my
opinions too seriously. (Not that there was any danger that you'd do
that, anyway.)
--
Cheers,
Herman Jurjus |
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| Jesse F. Hughes... |
| Posted: Tue Oct 13, 2009 6:07 am |
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Nam Nguyen <namducnguyen at (no spam) shaw.ca> writes:
[quote:2d2a5f9d81]Aatu Koskensilta wrote:
Rupert <rupertmccallum at (no spam) yahoo.com> writes:
Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive.
Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense. That aside, I was wondering specifically about the claim that the
well-ordering theorem itself is counter-intuitive, as alluded to in
e.g. the famous quip
The axiom of choice is obviously true, the well-ordering theorem
obviously false -- and who can tell of Zorn's lemma?
Just what intuitions are contradicted by the well-ordering theorem?
It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi),
where M(x) is the Major number of x and Pi and e are the 2 well known
transcendentals (please refer to the thread "Ttranscendental Goldbach
Conjecture"), since nobody would have a slightest intuition which one
assertion is the case.
[/quote:2d2a5f9d81]
So, your problem is with the law of excluded middle?
[quote:2d2a5f9d81]Since the well-ordering of reals implies one assertion over the other
is true, it's therefore counterintuitive.
[/quote:2d2a5f9d81]
That's not what I'd call counterintuitive. There are all sorts of
things that satisfy this. There is a nickel in my drawer, where I
can't see it. It is either heads up or tails up, but I don't have the
slightest intuition which one is the case. Do you find that situation
counterintuitive?
[quote:2d2a5f9d81]Put it differently, it's quite possible it's impossible to know
which inequality would hold - in all models of reals. It's therefore
counter- intuitive to talk about the well-ordering of the whole set
of reals.
[/quote:2d2a5f9d81]
Yes, I think it would be counterintuitive to speak of "the"
well-ordering, but surely people only speak of "a" well-ordering?
--
Jesse F. Hughes
"Why do the dirty villains always have to tie your hands *behind* ya?"
"That's what makes them villains."
--Adventures by Morse (old radio show) |
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| Daryl McCullough... |
| Posted: Tue Oct 13, 2009 7:32 am |
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Aatu Koskensilta says...
[quote:52d5fe4e65]Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense.
[/quote:52d5fe4e65]
I don't think what you are saying makes any sense. Bringing up
non-measurable sets is not a way to explain the counter-intuitiveness
of Banach-Tarski. It's a way of arguing that it *ISN'T* counter-intuitive
(because people don't have good intuitions about non-measurable sets).
The situation with Banach-Tarski to me is that (1) There is an informal,
intuitively true claim, (something along the lines of: If you
cut up a solid object into pieces, and you rearrange the pieces,
you'll get a new object that has the same volume as the original.
(2) There is an attempt to formalize the informal claim, by defining
what a "piece" might be, what "rearrange" means, etc.
(3) Then Banach-Tarski shows that the formal version is false.
You can argue that the informal claim is true, and that the
problem is that the formalization does not capture the informal
notion of "piece". But it is completely wrong to say that
the formalization depends on "baffling ideas" about non-measurable
sets. The formalization doesn't *MENTION* non-measurable sets.
Rather, the formalization just didn't specifically *RULE* *OUT*
non-measurable sets.
--
Daryl McCullough
Ithaca, NY |
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| WM... |
| Posted: Tue Oct 13, 2009 9:11 am |
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On 12 Okt., 20:16, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote:74a47252be]It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
[/quote:74a47252be]
Certainly. Well-ordering is equivalent to AC. But AC ...
"So, even though, for example, the Hausdorff-Banach-Tarski paradox
has
been called the most paradoxical result of the twentieth century,
classical mathematicians have to convince themselves that it is
natural, because it is a consequence of the Axiom of Choice, which
classical mathematicians are determined to uphold, because the Axiom
of Choice is required for important theorems which classical
mathematicians regard as intuitively natural."
[Henry Flynt: IS MATHEMATICS A SCIENTIFIC DISCIPLINE? (1994)]
http://www.henryflynt.org/studies_sci/mathsci.html
Regards, WM |
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| WM... |
| Posted: Tue Oct 13, 2009 9:46 am |
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On 12 Okt., 21:46, Nam Nguyen <namducngu... at (no spam) shaw.ca> wrote:
[quote:971036eade]Aatu Koskensilta wrote:
Nam Nguyen <namducngu... at (no spam) shaw.ca> writes:
Don't have a every details yet but it's possible the order be
relativistic over the (more absolute) existences of the reals, which
could give rise to paradoxes if we insist there's a "global" absolute
well order. Imho.
Whatever the merits of this, it appears to be a theoretical
(philosophical?) doctrine or hypothesis, not an intuition contradicted
by the well-ordering theorem.
It's only as much as philosophical as the current mathematical reasoning
based on the assumed knowledge of the naturals is. If you already
preconceived the truth of well-order of the reals isn't philosophically
based
[/quote:971036eade]
It is based on an erroneous proof.
Zeremlo's first proof of well ordering contains the sentence: "Wäre
nun m' das erste Element von M', welches von dem entsprechenden
Elemente m'' verschieden wäre, ..." meaning: If the gamma-set M' would
differ from the gamma-set M'' by the element m' for the first
time, ... [E. Zermelo: Beweis, daß jede Menge wohlgeordnet werden
kann, Math. Ann. 59 (1904) 514-516]
That implies: Zermelo's proof shows that the well-ordered set, e.g. R,
can be brought into an unbroken linear sequence, called gamma-set M',
were no element differs from the alternative sequence gamma-set M''.
There is no first element missing in one of the sets, and every
element has a precursor (otherwise it would not be sure whether the
element without precursor was the first element that could possible be
different in M' and M'').
Hence, Zermelo proves that R can be put into an unbroken sequence,
contradicting Cantor's proof.
Regards, WM |
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| WM... |
| Posted: Tue Oct 13, 2009 10:07 am |
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On 13 Okt., 02:24, Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote:855e58f3d6]Just what intuitions are contradicted by the well-ordering theorem?
[/quote:855e58f3d6]
Allegedly more numbers can be well-ordered than can be addressed or
named. What else, however, is ordering but addressing or naming
successively?
[quote:855e58f3d6]
--
Aatu Koskensilta (aatu.koskensi... at (no spam) uta.fi)
"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
[/quote:855e58f3d6]
There is no path to infinity, not even an endless one. [§ 123]
It isn't just impossible "for us men" to run through the natural
numbers one by one; it's impossible, it means nothing. […] you can’t
talk about all numbers, because there's no such thing as all numbers.
[§ 124]
Ludwig Wittgenstein, Philosophical Remarks
Regards, WM |
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| David C. Ullrich... |
| Posted: Wed Oct 14, 2009 8:17 am |
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On Mon, 12 Oct 2009 21:16:29 +0300, Aatu Koskensilta
<aatu.koskensilta at (no spam) uta.fi> wrote:
[quote:818fbcc3b3]
It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
[/quote:818fbcc3b3]
Come on. Surely it's clear that we need someone with a _less_
keen intuition to explain this.
So I'll step in. It contradicts the inuitively clear fact that there
is no well-ordering of R.
Seriously. You're not going to get a clear _mathematical_
answer to your question. I don't think that anyone's suggested
that the existence of a well-ordering of R is particularly
counterintuitive _to_ someone who's actually studied set
theory. But the quip about AC being obviously true while
WOT is clearly false simply _is_ the way it seems to many
people who haven't studied these things in any depth. Not
that I can explain why that should be, but I can give
evidence that it's so:
One sees people who don't see the need for AC as an
axiom since it's clearly true.
One sees people writing books where they carefully
state that this or that theorem requires AC, giving the
impression that they want to make it clear what parts
of the theory do and what parts do not depend on AC,
but then in the same book they prove theorems that
do depend on AC without acknowledging this, hence
I conjecture without being aware of it. (Eg books
on analysis treating, say, the Hahn-Banach theorem
and also giving a "careful" proof that a countable
union of countable sets is countable - a very smart
analyst down the hall simply didn't believe me
when I told him that that's not a theorem of ZF.) It
does seem to me that AC is so intuitively clear to
people that they use it all the time without realizing that
that's what they're doing (I for one am always very nervous
claiming that I have _not_ used AC anywhere in
some argument).
Otoh one sees the same people giving the impression
that the existence of an uncountable well-ordered
set requires AC. I'm not sure I've ever seen an
_explicit_ statement to this effect, but that's
certainly the impression one gets, especially when
the author seems to be trying to avoid AC when it's
not needed but uses it for this fact.
It does seem to me that for whatever reason people
_do_ use AC without even being aware of it, and
that simply is not true of WOT. Because AC simply
_is_ clear, while WOT is not.
That's just observation of what people seem to do,
hence it seems to me evidence of what their intuitions
actually _are_ - WOT contradicts the intuition that
R cannot be well-ordered, which is why we need AC,
well known to have counterintuitive consequences,
to prove it.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.) |
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| Herman Jurjus... |
| Posted: Wed Oct 14, 2009 9:52 am |
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David C. Ullrich wrote:
[quote:39b37ab608]On Mon, 12 Oct 2009 21:16:29 +0300, Aatu Koskensilta
aatu.koskensilta at (no spam) uta.fi> wrote:
It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
Come on. Surely it's clear that we need someone with a _less_
keen intuition to explain this.
So I'll step in. It contradicts the inuitively clear fact that there
is no well-ordering of R.
Seriously. You're not going to get a clear _mathematical_
answer to your question. I don't think that anyone's suggested
that the existence of a well-ordering of R is particularly
counterintuitive _to_ someone who's actually studied set
theory. But the quip about AC being obviously true while
WOT is clearly false simply _is_ the way it seems to many
people who haven't studied these things in any depth. Not
that I can explain why that should be, but I can give
evidence that it's so:
One sees people who don't see the need for AC as an
axiom since it's clearly true.
One sees people writing books where they carefully
state that this or that theorem requires AC, giving the
impression that they want to make it clear what parts
of the theory do and what parts do not depend on AC,
but then in the same book they prove theorems that
do depend on AC without acknowledging this, hence
I conjecture without being aware of it. (Eg books
on analysis treating, say, the Hahn-Banach theorem
and also giving a "careful" proof that a countable
union of countable sets is countable - a very smart
analyst down the hall simply didn't believe me
when I told him that that's not a theorem of ZF.)
[/quote:39b37ab608]
It always strikes me that people talk about applications of countable
choice or dependent choice as 'requiring AC'. I'd say that, when a
result can be proved with just CC and DC, it makes much more sense to
say that it does -not- require AC, even though perhaps it's not provable
from ZF alone. CC is sooo much weaker. (And much less controversial,
mathematically.)
[quote:39b37ab608]It
does seem to me that AC is so intuitively clear to
people that they use it all the time without realizing that
that's what they're doing
[/quote:39b37ab608]
Do you remember any cases where full AC was really required, instead of
just CC/DC?
[quote:39b37ab608](I for one am always very nervous
claiming that I have _not_ used AC anywhere in
some argument).
Otoh one sees the same people giving the impression
that the existence of an uncountable well-ordered
set requires AC. I'm not sure I've ever seen an
_explicit_ statement to this effect, but that's
certainly the impression one gets, especially when
the author seems to be trying to avoid AC when it's
not needed but uses it for this fact.
It does seem to me that for whatever reason people
_do_ use AC without even being aware of it, and
that simply is not true of WOT. Because AC simply
_is_ clear, while WOT is not.
That's just observation of what people seem to do,
hence it seems to me evidence of what their intuitions
actually _are_ - WOT contradicts the intuition that
R cannot be well-ordered, which is why we need AC,
well known to have counterintuitive consequences,
to prove it.
[/quote:39b37ab608]
Nothing snipped; spot on!
--
Cheers,
Herman Jurjus |
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