a < b implies 2^a < 2^b
quasi
(E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
Is (E) true in the standard model (S) of set theory?
Assuming the answer is no, let's analyze further ...
Let GCH denote the Generalized Continuum Hypothesis.
It's clear (is it?) that S + GCH => E.
Does S + E => GCH?
If not, and assuming E is independent of S, then S + E is
intermediate, strictly between S and S + GCH.
Disclosure (but probably obvious):
I know almost no set theory, thus every claim I make is somewhat
suspect. Corrections are welcome.
quasi
quasi
>Consider the following statement (E) ...
>
>(E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
>
>Is (E) true in the standard model (S) of set theory?
>
>Assuming the answer is no, let's analyze further ...
What I meant to say is:
Assuming E is independent of S, let's analyze further ...
Dave L. Renfro
> Consider the following statement (E) ...
>
> (E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
>
> Is (E) true in the standard model (S) of set theory?
I'm pretty sure (E) is an independent statement.
In fact, I believe even the special case in which
a = aleph_0 and b = aleph_1 is independent, as this
special case is sometimes called "Lusin's Hypothesis"
(because Lusin first drew attention to it, sometime
in the 1920s I think), and I'm pretty sure Lusin's
Hypothesis has been proved independent (probably back
in the late 1960s).
As for the other things you asked about (not copied above),
I don't know enough set theory to answer. I could probably
find an answer looking through some books I have at home,
but I'm not there now and I don't have any of my references
with me where I'm at.
Dave L. Renfro
Dave L. Renfro
Regarding the cardinal "function" b --> 2^b, I think
pretty much anything is possible as long as certain
monotonicity features are preserved (one of which comes
from Konig's cardinal inequality theorem and involves
the cofinality of b). Some of the comments in the
following thread may be of interest to you.
http://groups.google.com/group/sci.math/browse_frm/thread/deb3966b411d05c0
Dave L. Renfro
Butch Malahide
> Consider the following statement (E) ...
>
> (E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
>
> Is (E) true in the standard model (S) of set theory?
The standard "model" of ZFC is the class of all sets, usually denoted
by V.
> Assuming the answer is no, let's analyze further ...
>
> Let GCH denote the Generalized Continuum Hypothesis.
>
> It's clear (is it?) that S + GCH => E.
But "V + GCH" makes no sense. Perhaps by S you meant ZFC?
> Does S + E => GCH?
ZFC + E does not imply GCH; e.g., it could be that 2^{aleph_alpha} =
aleph_{alpha + 2} for every alpha.
> If not, and assuming E is independent of S, then S + E is
> intermediate, strictly between S and S + GCH.
Correct, assuming "S" means "ZFC".
> Disclosure (but probably obvious):
>
> I know almost no set theory, thus every claim I make is somewhat
> suspect. Corrections are welcome.
Same here.
Han de Bruijn
> Consider the following statement (E) ...
>
> (E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
>
> Is (E) true in the standard model (S) of set theory?
Who cares about any "standard model"? If a and b are whatever numbers,
then (a < b, then 2^a < 2^b) is simply true.
Han de Bruijn
quasi
I'm almost certain that you're wrong, but if you think you can prove
it, let's see a proof. Cardinal numbers are not obliged to match your
intuition as to their algebraic relationships.
Here's what is true:
(1) For all cardinals a,b
2^a < 2^b implies a < b
(2) For all cardinals a,b,
a < b implies 2^a <= 2^b
(3) If we assume the standard axioms of set theory, _plus_ the
Generalized Continuum Hypothesis, then
For all cardinals a,b,
a < b implies 2^a < 2^b.
But based on the discussion so far, it seems likely that the statement
"For all cardinals a,b,
a < b implies 2^a < 2^b"
is independent of the standard axioms of set theory.
quasi
Han de Bruijn
Why don't you say that your default cardinal is an _infinite_ cardinal.
Because, for finite cardinals, (a < b, then 2^a < 2^b) is simply true.
I don't understand the choice of your default, though, because infinite
cardinals IMO do not exist and hence aren't worth considering. Why solve
a problem that simply isn't there in the whole universe?
Han de Bruijn
Derek Holt
> Consider the following statement (E) ...
>
> (E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b.
>
> Is (E) true in the standard model (S) of set theory?
It's independent of the axioms of the standard model. I once made use
of that implication in a published paper, and somebody else promptly
published a corrected version proving that the result I had proved was
also independent of the standard model. It is one of the craziest
aspects of using numbers of citations as a measure of research quality
that papers containing errors will often get more citations (pointing
out the error) than flawless papers.
I have also come across a proof in a standard textbook of the theorem
If F1 and F2 are isomorphic free groups with free bases X1 and X2,
then |X1|=|X2|
which considers homomorphisms of F1 and F2 onto the group of order 2,
deduces that 2^|X1| = 2^|X2| and concludes that |X1|=|X2|. Of course
there are alternative correct proofs of this result.
Derek Holt.
quasi
<Han.de...@DTO.TUDelft.NL> wrote:
Well, the standard model of set theory _has_ infinite cardinals. In
any mathematical discussion, unless specified otherwise, the standard
model is assumed, by default. Besides, in the body of my original
post, I even specified the standard model.
>Because, for finite cardinals, (a < b, then 2^a < 2^b) is simply true.
Yes, of course.
>I don't understand the choice of your default, though, because infinite
>cardinals IMO do not exist and hence aren't worth considering.
Ok, but that's a non-standard view. I'm not saying it's wrong, but
it's certainly non-standard. For my question, I was using the standard
model, and as you can see, the question does not have such an easy
answer in the standard model.
>Why solve a problem that simply isn't there in the whole universe?
Because it's interesting.
Because it's challenging.
Because it's fun.
That's enough for me.
But in any case, many of these theories about things that you think
don't exist give results about things that you surely believe _do_
exist.
As an analogy, use of sqrt(-1) resolves some difficult problems about
the real numbers. Thus, we should be willing to think outside the box
of "reality" if only to better understand reality.
quasi
Tonico
************************************************
In some strange way it is a little reassuring to see that people,
including trolls and cranks and their nonsense, continue in their
paths
As usual Han, you say there is no infinite cardinals in the universe,
and as usual you're answered: who gives a damn what is or what isn't
in YOUR universe?
Anyway you'd have problems to see a number 7 or a number -14.67
walking down the street anywhere in your universe, so why won't you
cuddle in your universe and let us mathematicians alone with our
fantasies?
We enjoy infinite cardinals and all the theory around, it is a great
mental exercise and a rather pleasant one: who do you think you and
the non-mathematicians ones like you (Mueckenheim, Orlow, Zick and
other rather sad examples of the lousy mathematical education in big
parts of Europe after WWII) are to tell US mathematicians what can we
or can't we deal with?
The only tiny, subtle basis to your ridiculous and absurd whinning
would be to prove it is a self-contradicting theory, and you people
haven't shown anything even close to this, most probably because of
the simple reason you are not mathematicians and you do don't have,
usually, the slightest idea what you're talking about.
So...freak off and let mathematicians deal with mathematics and our
demons, fairies, dreams and fantasies. In the meantime you technicians
use what we develop for you.
Regards
Tonio
user923005
It is certainly true for real-valued a anb b, but is it true for all
complex a and b?
(Just asking, because I am not certain of the answer or even if (a <
b) is properly defined by their magnitudes or some other measure)
> I don't understand the choice of your default, though, because infinite
> cardinals IMO do not exist and hence aren't worth considering. Why solve
> a problem that simply isn't there in the whole universe?
Because it is interesting?
quasi
Right.
> but is it true for all complex a and b?
>(Just asking, because I am not certain of the answer or even if (a <
>b) is properly defined by their magnitudes or some other measure)
For complex numbers, the statement
"a < b implies 2^a < 2^b"
is not meaningful.
The issue is that the field of complex numbers is not an ordered
field.
For example, how does i compare to 0?
If i > 0, then i^2 > 0, contradiction.
If i = 0, then i^2 = 0, contradiction.
If i < 0, then i^2 > 0, contradiction.
Hence there is no comparison.
Of course, you _can_ compare the magnitudes, but that won't give an
ordering that properly respects the arithmetic operations.
However, for the question I asked in my original post, the symbols a,b
are infinite cardinal numbers, hence are neither real nor complex.
Infinite cardinals provide a way to compare the sizes of infinite
sets. If the concept is unfamiliar, try looking up "cardinal numbers".
quasi
Han de Bruijn
> But in any case, many of these theories about things that you think
> don't exist give results about things that you surely believe _do_
> exist.
>
> As an analogy, use of sqrt(-1) resolves some difficult problems about
> the real numbers. Thus, we should be willing to think outside the box
> of "reality" if only to better understand reality.
Sure. Sqrt(-1) = i = (0,1) , which is a vector in _real_ 2-D space. Now
define the elementary operations (e.g. multiplication) for such vectors
and you're done.
As a theoretical physicist, I have a decent (non naive) understanding of
reality. I know that (actual) infinity is not in there. But I'll respect
your choice.
Han de Bruijn
Han de Bruijn
And in YOURs. For the simple reason that there is only ONE universe we
are all dealing with, whether you like it or not.
> Anyway you'd have problems to see a number 7 or a number -14.67
> walking down the street anywhere in your universe, so why won't you
> cuddle in your universe and let us mathematicians alone with our
> fantasies?
Is mathematics an activity of economic value? Is it rewarded? If yes,
then feel the responsability of producing something else than fantasy.
> We enjoy infinite cardinals and all the theory around, it is a great
> mental exercise and a rather pleasant one: who do you think you and
> the non-mathematicians ones like you (Mueckenheim, Orlow, Zick and
> other rather sad examples of the lousy mathematical education in big
> parts of Europe after WWII) are to tell US mathematicians what can we
> or can't we deal with?
Yeah, yeah, I _enjoy_ a lot of things I am _not_ payed for. I mean: do
the enjoying in your free time. And don't ask money for it.
> The only tiny, subtle basis to your ridiculous and absurd whinning
> would be to prove it is a self-contradicting theory, and you people
> haven't shown anything even close to this, most probably because of
> the simple reason you are not mathematicians and you do don't have,
> usually, the slightest idea what you're talking about.
Oh yes, the usual blather .. But deep in your heart, you know better,
don't you?
> So...freak off and let mathematicians deal with mathematics and our
> demons, fairies, dreams and fantasies. In the meantime you technicians
> use what we develop for you.
I fear that about 98 percent of what you produce is not usable at all.
Han de Bruijn
Han de Bruijn
>>Why don't you say that your default cardinal is an _infinite_ cardinal.
>>Because, for finite cardinals, (a < b, then 2^a < 2^b) is simply true.
>
> It is certainly true for real-valued a anb b, but is it true for all
> complex a and b?
> (Just asking, because I am not certain of the answer or even if (a <
> b) is properly defined by their magnitudes or some other measure)
Finite cardinals are just naturals and for naturals it's certainly true.
Complex numbers have no intrinsic order, so not (a < b, then 2^a < 2^b)
if a and b are complex, because < is not defined for complex numbers.
>>I don't understand the choice of your default, though, because infinite
>>cardinals IMO do not exist and hence aren't worth considering. Why solve
>>a problem that simply isn't there in the whole universe?
>
> Because it is interesting?
Re-define interesting?
Han de Bruijn
Jesse F. Hughes
What you don't understand is the universal quantifier. If one asks
"Is it true that if a and b are cardinals,...", they *mean* "Is the
following true for *every* pair of cardinals?"
You seem to think they mean, "Is this true for 'default' cardinals?"
but this question is meaningless, since there is no clear notion of
default cardinals.
--
"So why talk [about my factoring method] out on Usenet? Because it's a
highly public place so I'm unlikely to disappear[...] You people are
my protection. [...] You may be what's keeping me free and walking out
in the open air." -- James S. Harris, theory guy on the edge.
Jesse F. Hughes
> As a theoretical physicist, I have a decent (non naive) understanding of
> reality. I know that (actual) infinity is not in there.
Sure. A four year degree in Theoretical Physics gives one clear
insight into such questions. I am sure we all recognize your
authority here.
--
Jesse F. Hughes
"We will run this with the same kind of openness that we've run
Windows." Steve Ballmer, speaking about MS's new ".Net" project.
quasi
<Han.de...@DTO.TUDelft.NL> wrote:
>Is mathematics an activity of economic value? Is it rewarded? If yes,
>then feel the responsability of producing something else than fantasy.
Hey, we're artists, not engineers.
Mathematics can be regarded as "the art of thinking abstractly".
Sure it's been applied, and sure it uses the "scientific method", but
at heart, it's art, not science.
Are you saying that the quest to achieve high levels of abstract
thought has no value?
Are you saying "art" has no value to society.
Thus, artists, musicians, novelists, screenwriters, actors,
professional athletes, chess players, philosophers -- all have no
value? Or they do have value, and deserve to be paid, but not
mathematicians?
One could equally ask whether there's much economic value in what some
theoretical physicists do. For that matter, one can debate whether
there's much economic value in what theoretical economists do.
The art of pure thought has been revered since ancient times. Its
value is in its insights and in the inspiration it offers. Sometimes
it gets applied, sometime not. No one cares. It's beautiful. Luckily,
most key decision makers acknowledge the underlying value, otherwise
support would surely be cut off.
quasi
tommy1729
true for reals.
true for the absolute value of complex numbers.
however for more complicated numbers , this statement is not so obvious , perhaps even false (matrices , nilpotence , octions etc )
however quasi asked about cardinals which behave different than reals from the viewpoint of most set theories and certainly the standard set theory.
there is no reason to be analogue to reals.
in fact it seems very inconsistant since e.g.
the reals between 0 and 1 have the same cardinality as the reals between 0 and 2.
the ones who read my posts in the past will know i dont support the axioms of standard set theory.
so asking if a statement is consistant with respect to standard set theory and its axioms sounds to me like asking
axioms : 2+2 =5 ; 1+7 =19 ; 2^3 =17
is 1 + 1 = 3 ??
however (E) should be correct by cantors theorem ;
since 2^ implies getting a higher cardinality
<=> a < b <-> 2^a < 2^b
but in my viewpoint Han de Bruijn and Cantor made the same mistake : making analogues to (finite) reals.
and it has been shown that the truth of cantor only depends on the axioms chosen and is thus unprovable ;
so cantor has no way of defending that ( silly ) analogue.
despite cantor's words that cardinals are very different from reals , he ironicly threats them somewhat like them.
not the numbers themselves , but their orders are threated like 2log (n).
and thats his biggest mistake.
i have 2 set theories , and will answer quasi's question with respect to them.
these 2 are the only ones that make sence according to me.
i will list the conclusions of each before answering (E).
both use N = aleph_0 and R = aleph_1
1) 2^N = R ; R^2 = R ; 2^R = R ; R^R = R ;
a^b = b^a for a;b > N ; R = max
gives N < R and 2^N = 2^R
so (E) fails here
2) 2^N = R ; R^2 > R ; 2^R = R^2 ; R^R > 2^R ;
a^b = b^a for a;b > N ; no max
(E) is correct here.
>
> Han de Bruijn
>
im sorry to somewhat disagree Han , yet regards to you.
tommy1729
victor_me...@yahoo.co.uk
> Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> writes:
>
> What you don't understand is the universal quantifier. If one asks
> "Is it true that if a and b are cardinals,...", they *mean* "Is the
> following true for *every* pair of cardinals?"
>
> You seem to think they mean, "Is this true for 'default' cardinals?"
> but this question is meaningless, since there is no clear notion of
> default cardinals.
And of course the notion of "default cardinals" does not appear in
the OP's posting. Naturally this notion is a face-saving device of
de Bruijn to compensate for his misreading not only of the
universal quantifier but also of the OP's word "cardinal".
Alas his face-saving has also led to more and more hysterical
postings culminating in his right-wing diatribe about the
"economic value" of mathematics. His repressive political agenda
is showing, and it is not a pretty sight.
Victor Meldrew
"I don't believe it!"
tommy1729
no quasi , han was defending against tonico.
dont take it personal.
you and han actually agree that set theorists dont contribute much to science , rather to art.
whereas tonico implied that Han is just dumb , doesnt know what he is saying , and mathematicians (set theorists ; he ? ) contribute to science.
But Han is not dumb , and set theory has no practical applications.
However i dont totally agree with Han either , see my other post here.
Han however defended himself pretty well.
Dont mistake his defense with an attack on mathematicians in general.
Of course some set theorists or tonico would like that to get implied ; so that Han does look like a crank who hates mathematicians , but those tricks dont work on me.
And i hope neither on you.
regards
tommy1729
Han de Bruijn
> On Wed, 09 Jan 2008 13:48:38 +0100, Han de Bruijn
> <Han.de...@DTO.TUDelft.NL> wrote:
>
>>Is mathematics an activity of economic value? Is it rewarded? If yes,
>>then feel the responsability of producing something else than fantasy.
>
> Hey, we're artists, not engineers.
>
> Mathematics can be regarded as "the art of thinking abstractly".
>
> Sure it's been applied, and sure it uses the "scientific method", but
> at heart, it's art, not science.
>
> Are you saying that the quest to achieve high levels of abstract
> thought has no value?
>
> Are you saying "art" has no value to society.
>
> Thus, artists, musicians, novelists, screenwriters, actors,
> professional athletes, chess players, philosophers -- all have no
> value? Or they do have value, and deserve to be paid, but not
> mathematicians?
There is a _crucial difference_. Artists, musicians, novelists, actors,
screenwriters, professional athletes, chess players, they all entertain
_other_ people, rather than themselves. While mathematicians tend to be
quite boring for virtually anybody else than themselves. If mathematics
intends to be an art, it should be enjoyable in the first place, and by
this I mean: not only for mathematicians. But I said: if ..
> One could equally ask whether there's much economic value in what some
> theoretical physicists do. For that matter, one can debate whether
> there's much economic value in what theoretical economists do.
>
> The art of pure thought has been revered since ancient times. Its
> value is in its insights and in the inspiration it offers. Sometimes
> it gets applied, sometime not. No one cares. It's beautiful. Luckily,
> most key decision makers acknowledge the underlying value, otherwise
> support would surely be cut off.
Han de Bruijn
Gonçalo Rodrigues
<Han.de...@DTO.TUDelft.NL> fed this fish to the penguins:
>quasi wrote:
>
>> But in any case, many of these theories about things that you think
>> don't exist give results about things that you surely believe _do_
>> exist.
>>
>> As an analogy, use of sqrt(-1) resolves some difficult problems about
>> the real numbers. Thus, we should be willing to think outside the box
>> of "reality" if only to better understand reality.
>
>Sure. Sqrt(-1) = i = (0,1) , which is a vector in _real_ 2-D space. Now
>define the elementary operations (e.g. multiplication) for such vectors
>and you're done.
>
You are totally missing the point. The point is that all successful
theories have had the need to recourse to *abstract* mathematical
tools that are removed away from our general experience. Quasi
mentioned the complex numbers, but we could add the Minkowski space
for special relativity or the curvature tensor for General relativity,
physical quantities as operators in Quantum Mechanics with the their
spectrum as the allowed measurable values, the covering group SU(2) to
treat the spin of a particle, etc. The successful formulation of these
physical theories has had an absolute need for concepts that are not
borne from ordinary experience, but by *mathematical abstraction*, or,
to repeat Quasi, in order to probe and explain what you see, you had
to imagine the unseen.
>As a theoretical physicist, I have a decent (non naive) understanding of
>reality. I know that (actual) infinity is not in there. But I'll respect
>your choice.
>
A theoretical physicist?
I have by me the book "Quantum Mechanics", second edition, by E.
Merzbacher, from which I learned the subject a few years ago (along
with some others like the "Quantum Mechanics" by L. Landau). Turning
to chapter 8. entitled "Principles of Wave Mechanics", on the start of
the second section we can read:
"We have learned that every physical quantity F can be represented by
a linear operator, which for convenience is denoted by the same
letter, F."
If we go on we also learn that these linear operators are also
self-adjoint - so that their spectrum is real. Self-adjointness is
tied to existence of an inner product in the space. Turning to chapter
14 we would learn that the spaces where these operators act are
Hilbert spaces, which in general are separable but *not necessarily*
of finite dimension. A separable infinite-dimensional Hilbert space is
as infinite as infinite gets.
We could multiply the examples infinitely (pun intended) drawing from
all sources from Classical Mechanics, Special and General Relativity,
etc. The fact is, is that all the successful theories that model our
universe use highly infinitary mathematical tools.
But do not let these simple facts spoil your fantasies.
Best regards,
G. Rodrigues
Gonçalo Rodrigues
<Han.de...@DTO.TUDelft.NL> fed this fish to the penguins:
>quasi wrote:
>
>> On Wed, 09 Jan 2008 13:48:38 +0100, Han de Bruijn
>> <Han.de...@DTO.TUDelft.NL> wrote:
>>
>>>Is mathematics an activity of economic value? Is it rewarded? If yes,
>>>then feel the responsability of producing something else than fantasy.
>>
>> Hey, we're artists, not engineers.
>>
>> Mathematics can be regarded as "the art of thinking abstractly".
>>
>> Sure it's been applied, and sure it uses the "scientific method", but
>> at heart, it's art, not science.
>>
>> Are you saying that the quest to achieve high levels of abstract
>> thought has no value?
>>
>> Are you saying "art" has no value to society.
>>
>> Thus, artists, musicians, novelists, screenwriters, actors,
>> professional athletes, chess players, philosophers -- all have no
>> value? Or they do have value, and deserve to be paid, but not
>> mathematicians?
>
>There is a _crucial difference_. Artists, musicians, novelists, actors,
>screenwriters, professional athletes, chess players, they all entertain
>_other_ people, rather than themselves. While mathematicians tend to be
>quite boring for virtually anybody else than themselves. If mathematics
>intends to be an art, it should be enjoyable in the first place, and by
>this I mean: not only for mathematicians. But I said: if ..
>
You yourself do not *understand* mathematics, you do not enjoy it, and
from your own personal bias and prejudice, you leap to the conclusion
that no one but mathematicians enjoys mathematics. Wow.
Never mind reducing the function of art to the "entertainment" of
others. This is the typical inanity spewed by a complete inability to
make fine distinctions, and joining in the same phrase, say, the
Divine Comedy, and the latest Batman comic strip.
By God, what an arrogant idiot you are.
G. Rodrigues
Derek Holt
Is that US as in "you and me" (if so, why is it capitalized?), or US
as in USA (comparing it with Europe)?
Derek Holt.
Dave L. Renfro
>> Consider the following statement (E) ...
>>
>> (E): If a,b are cardinal numbers such that a < b,
>> then 2^a < 2^b.
>>
>> Is (E) true in the standard model (S) of set theory?
Dave L. Renfro wrote (in part):
> I'm pretty sure (E) is an independent statement.
> In fact, I believe even the special case in which
> a = aleph_0 and b = aleph_1 is independent, as this
> special case is sometimes called "Lusin's Hypothesis"
> (because Lusin first drew attention to it, sometime
> in the 1920s I think), and I'm pretty sure Lusin's
> Hypothesis has been proved independent (probably back
> in the late 1960s).
Actually, this was proved in the Summer or Fall
of 1963, by Robert Solovay.
I looked over some notes I have at home on this
topic and here's a brief summary. Incidentally,
when I say "we can have", this roughly means
there is a model of the ZFC axioms in which the
statement is true. Some of what follows came from
expository papers (Gregory H. Moore has written
a couple), some from paper abstracts or summaries
or Math. Reviews, and some from Solovay himself
(who wrote a nice letter to me back in Summer 2001
in reply to some questions I asked him).
By the way, I know very little about the technical
aspects of these things, but I do know that many of
these results have more precise versions that have to
do with how "nice" the model is in certain ways.
In the Spring of 1963 (it may have been the month
of May), Paul Cohen gave two talks at Princeton
on his results. The first was for experts and
Robert Solovay, who at the time was a topologist
visiting from Berkeley, didn't attend, but Solovay
did attend the second talk that was for a more
general audience. At the talks Cohen outlined his
results, one of which was that 2^(aleph_0) can be
arbitrarily large. Cohen mentioned that he didn't
know, for example, whether 2^(aleph_n) being equal
to aleph_(n+1) for n = 1, 2, 3, 4, 5 implies
that 2^(aleph_6) = aleph_7. [I think his actual
question was a bit more precise, something along
the lines of if aleph_6 Cohen reals are added to
a transitive model of ZFC + GCH, then in the new
model does the set of real numbers have cardinality
aleph_6 or cardinality aleph_7. And don't ask me
what this means!]
Solovay decided to work on this problem and was
able to solve it later that year. Solovay's announcement
is in a 1963 issue of the Notices of the AMS and had
the catchy title "2^aleph_0 can be anything it ought
to be". In this announcement (it later also appeared
as a one-page announcement on p. 435 of "The Theory
of Models", North-Holland, 1965; this is a publication
for a conference held at Berkeley in 1963), Solovay
states these 3 results ('cf' is 'cofinality'; see the
end of this post for an explanation):
1. Let be be a cardinal with uncountable cofinality,
i.e. cf(b) > aleph_0. Then we can have 2^(aleph_0) = b.
By the time Solovay announced this, I believe Cohen
had also proved it. Note this is a bit more precise
than "2^(aleph_0) can be arbitrarily large".
2. Let b' = cf(b') < cf(b). Then we can have
2^(b') = b and
(for all a)(aleph_a < b' implies 2^(aleph_a) = aleph_(a+1)).
3. Let 0 <= n_1 <= n_2 <= ... <= n_k be finite cardinals.
Then we can have 2^(aleph_j) = aleph_(n_j)
for each j = 1, 2, ..., k.
Note: #3 implies that what I called "Lusin's Hypothesis"
(yesterday) can hold. By the way, "Lusin's [Luzin's]
Hypothsis" is also called the "second continuum hypothesis".
Solovay's method in #3 only allowed him to control
the values of a finite number of cardinals. Although
the published statement was for alephs with finite
ordinal subscripts, I believe Solovay's method worked
for any finite number of regular cardinals. However,
I'm not very sure about this point. In the following
year (1964), William B. Easton managed to control the
powers (with base 2) of all the regular cardinals at
once. This was Easton's 1964 Ph.D. Dissertation at
Princeton, under Alonzo Church. For some reason it
wasn't published until 1970 [in Annals of Mathematical
Logic, volume 1, pp. 139-178].
Easton's main result was that, given any "function" f
from the regular ordinals to the cardinals that satisfies
certain very general conditions, we can have 2^b = f(b)
for all regular cardinals b. There are two conditions on f:
A. a < b implies f(a) <= f(b).
B. For all b, we have b < cf( f(b) ).
Condition A obviously must hold, and condition B is
a restriction that arises from Konig's strict inequality
involving transfinite sums and products. What happened
is that at the 1904 International Congress of Mathematicians,
J. Konig gave a talk in which he used some cardinal
computations to disprove the continuum hypothesis.
In fact, I think he went much further, asserting
that the cardinality of the continuum cannot be equal
to aleph_b for any ordinal b. It was not long afterwards
(at most a few days, I think) that an error was found
in Konig's "proof", but one of the things that survived
the corrections to what Konig presented is the result that
the cardinality of the continuum cannot have countable
cofinality. Thus, 2^(aleph_0) cannot equal aleph_w or
aleph_(w^2 + w) or aleph_(epsilon_0), among other things.
The situation for singular cardinals is, from what
I understand, much more involved and problematic.
I think Jack Silver was the first to come up with
some definitive results showing that the kind of
behavior Solovay and Easton obtained for the possible
behavior of the function 2^b for regular cardinals b
does not hold for singular cardinals. [Easton's methods
simply didn't allow singular cardinals to be treated,
which of course still allows for the possibility
that some other method might do the trick. Silver
showed this doesn't happen, however.] An example of
the result that Silver proved (announced in the summer
of 1974, I think) is that if [in certain "nice" models
of ZFC] 2^(aleph_b) = aleph_(b+1) for each ordinal
b < omega_1, then we must have 2^(aleph_omega_1)
equal to aleph_(omega_1 + 1). In fact, Silver proved
this for any singular cardinal with uncountable
cofinality replacing aleph_omega_1.
COFINALITY OF A CARDINAL -- This is the least limit
ordinal through which the cardinal can be reached
(as a limit of) by using a transfinite ordinal sequence
of smaller cardinals. The cofinality of a cardinal is
itself a cardinal.
Examples.
The cofinality of each of these is omega (i.e. w, or w_0):
(1) aleph_0, (2) aleph_omega, (3) the smallest cardinal b
such that aleph_b = b.
(1) aleph_0 = sup{1, 2, 3, ...}
(2) aleph_omega = sup{aleph_1, aleph_2, aleph_3, ...}
(3) sup{aleph_w, aleph_(aleph_w), aleph_(aleph_(aleph_w))), ...}
The cofinality of any cardinal of the form aleph_(b+1) for
some ordinal b is aleph_(b+1).
A cardinal b is regular if cf(b) = b and singular
if cf(b) < b. [It is not possible to have cf(b) > b.]
Dave L. Renfro
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
Not without appropriate definitions of "2", "a", "b", "," "^" and "then".
Virgil
Then HdB's "whole universe" is too small.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
We mathematicians have the same power to declare limits on the "reality"
of theoretical physicists as they have to declare limits on ours.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> > ************************************************
> >
> > In some strange way it is a little reassuring to see that people,
> > including trolls and cranks and their nonsense, continue in their
> > paths
> >
> > As usual Han, you say there is no infinite cardinals in the universe,
> > and as usual you're answered: who gives a damn what is or what isn't
> > in YOUR universe?
>
> And in YOURs. For the simple reason that there is only ONE universe we
> are all dealing with, whether you like it or not.
>
> > Anyway you'd have problems to see a number 7 or a number -14.67
> > walking down the street anywhere in your universe, so why won't you
> > cuddle in your universe and let us mathematicians alone with our
> > fantasies?
>
> Is mathematics an activity of economic value? Is it rewarded? If yes,
> then feel the responsability of producing something else than fantasy.
If your test of things mathematical is that they must be useful, by that
same argument we must discount all art, music, and anything else which
appeals only to the esthetic.
>
> > We enjoy infinite cardinals and all the theory around, it is a great
> > mental exercise and a rather pleasant one: who do you think you and
> > the non-mathematicians ones like you (Mueckenheim, Orlow, Zick and
> > other rather sad examples of the lousy mathematical education in big
> > parts of Europe after WWII) are to tell US mathematicians what can we
> > or can't we deal with?
>
> Yeah, yeah, I _enjoy_ a lot of things I am _not_ payed for. I mean: do
> the enjoying in your free time. And don't ask money for it.
So don't ask money for music, or movies, etc.?
>
> > The only tiny, subtle basis to your ridiculous and absurd whinning
> > would be to prove it is a self-contradicting theory, and you people
> > haven't shown anything even close to this, most probably because of
> > the simple reason you are not mathematicians and you do don't have,
> > usually, the slightest idea what you're talking about.
>
> Oh yes, the usual blather .. But deep in your heart, you know better,
> don't you?
We certainly know better than HdB about what he chooses to despise.
>
> > So...freak off and let mathematicians deal with mathematics and our
> > demons, fairies, dreams and fantasies. In the meantime you technicians
> > use what we develop for you.
>
> I fear that about 98 percent of what you produce is not usable at all.
Not by the likes of HdB at any rate. But that is his loss, not ours.
Such pedestrianism limits him.
>
> Han de Bruijn
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
Of interest to someone other than HdB.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> quasi wrote:
>
> > On Wed, 09 Jan 2008 13:48:38 +0100, Han de Bruijn
> > <Han.de...@DTO.TUDelft.NL> wrote:
> >
> >>Is mathematics an activity of economic value? Is it rewarded? If yes,
> >>then feel the responsability of producing something else than fantasy.
> >
> > Hey, we're artists, not engineers.
> >
> > Mathematics can be regarded as "the art of thinking abstractly".
> >
> > Sure it's been applied, and sure it uses the "scientific method", but
> > at heart, it's art, not science.
> >
> > Are you saying that the quest to achieve high levels of abstract
> > thought has no value?
> >
> > Are you saying "art" has no value to society.
> >
> > Thus, artists, musicians, novelists, screenwriters, actors,
> > professional athletes, chess players, philosophers -- all have no
> > value? Or they do have value, and deserve to be paid, but not
> > mathematicians?
>
> There is a _crucial difference_. Artists, musicians, novelists, actors,
> screenwriters, professional athletes, chess players, they all entertain
> _other_ people, rather than themselves. While mathematicians tend to be
> quite boring for virtually anybody else than themselves. If mathematics
> intends to be an art, it should be enjoyable in the first place, and by
> this I mean: not only for mathematicians. But I said: if ..
That HdB is not entertained by something, does not mean no one is.
Grand Opera entertains relatively few people other than those who
produce it, so the issue must be if there are ANY non-mathematicians who
are entertained by the mathematics that does not entertain HdB, then it
is justifiable.
lwa...@lausd.net
> In article <190d6$4784c2a6$82a1e228$21...@news2.tudelft.nl>,
> > Is mathematics an activity of economic value? Is it rewarded? If yes,
> > then feel the responsability of producing something else than fantasy.
> If your test of things mathematical is that they must be useful, by that
> same argument we must discount all art, music, and anything else which
> appeals only to the esthetic.
And here we go again with that buzzword "useful."
Both standard and nonstandard mathematicians accuse
the other side's theory of not being "useful."
And what makes a theory "useful," anyway? The word
"useful," apparently, is redefined so that the
person employing that word (whether a proponent or
an opponent of standard theory) can say that his
own theory is useful and the other's isn't.
Virgil and HdB are two of the biggest culprits when
it comes to proclaiming one's own theory as useful
and each other's as not useful.
BTW, since the OP (quasi) specifically asked about
standard set theory, only standard theory like ZFC,
not a nonstandard theory like ZF-Infinity+~Infinity
need be considered.
IMO, "useful" is the most _useless_ word there is
when it comes to describing mathematics.
lwa...@lausd.net
> i have 2 set theories , and will answer quasi's question with respect to them.
> these 2 are the only ones that make sence according to me.
> i will list the conclusions of each before answering (E).
> both use N = aleph_0 and R = aleph_1
> 1) 2^N = R ; R^2 = R ; 2^R = R ; R^R = R ;
> a^b = b^a for a;b > N ; R = max
> gives N < R and 2^N = 2^R
> so (E) fails here
OK, I know this is Pocket Set Theory/TST.
> 2) 2^N = R ; R^2 > R ; 2^R = R^2 ; R^R > 2^R ;
> a^b = b^a for a;b > N ; no max
> (E) is correct here.
But what theory of tommy1729's is this?
lwa...@lausd.net
> On Jan 9, 5:28 am, tommy1729 <tommy1...@gmail.com> wrote:
> > 2) 2^N = R ; R^2 > R ; 2^R = R^2 ; R^R > 2^R ;
> > a^b = b^a for a;b > N ; no max
Rereading this, one line stands out at me:
> > a^b = b^a for a;b > N
So exponentiation has been declared commutative for all cardinals
greater than aleph_0.
In the spirit of the OP (quasi), one might wonder whether there is a
model of a standard theory (such as ZFC) in which exponentiation
is commutative for uncountable cardinals.
But we see that this is impossible. Suppose a=C (the cardinality of
the continuum) and b=2^C. The we obtain:
b^a = (2^C)^C = 2^(C*C) = 2^C
a^b = C^(2^C) >= 2^(2^C) > 2^C
So b^a > a^b, regardless of whether (G)CH is true in our model. So
clearly 2) is not true in any standard set theory.
I'm wondering what sort of set theory tommy1729 is imagining such
that uncountable exponentiation is commutative. Perhaps one
could imagine a sort of Pocket Set Theory in which there are three
infinite cardinalities, aleph_0, aleph_1, and aleph_2, where all
proper classes have cardinality aleph_2. Then naturally we would
have aleph_1^aleph_2 = aleph_2^aleph_1 = aleph_2, so that
uncountable exponentiation would be commutative. But then
tommy1729 writes "no max," so we can't have a PST with a
maximum cardinality reserved for proper classes.
Some of the other formulae tommy1729 has written, such as
"R^2 > R" and "R^R > 2^R," are embraced by those opponents of
ZFC who believe that proper subsets should have strictly smaller
cardinality than the original set (such as TO, RF, etc.). But none of
them have ever suggested that exponentiation be commutative. The
only proponent of commutative exponentiation here is tommy1729.
Aatu Koskensilta
> despite cantor's words that cardinals are very different from reals , he
> ironicly threats them somewhat like them.
Cantor's horrible threats are indeed famous. It's good to see you're still
at the business of tearing down rotten orthodoxy.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> There is a _crucial difference_. Artists, musicians, novelists, actors,
> screenwriters, professional athletes, chess players, they all entertain
> _other_ people, rather than themselves.
There are many fields where artists arguably entertain only each other.
Han.de...@dto.tudelft.nl
> On Wed, 09 Jan 2008 15:49:42 +0100,Han de Bruijn
> <Han.deBru...@DTO.TUDelft.NL> fed this fish to the penguins:
Talking about "fine distinctions" (see below), could it be possible
that the words "rather", "virtually", "If", "intends to be", "should"
add somewhat more colour to your black and white interpretation?
> Never mind reducing the function of art to the "entertainment" of
> others. This is the typical inanity spewed by a complete inability to
> make fine distinctions, and joining in the same phrase, say, the
> Divine Comedy, and the latest Batman comic strip.
Extract all "fine distinctions" from someone's comments and nobody can
deny anymore that you are right. Sure.
> By God, what an arrogant idiot you are.
Preferring the personal attack, instead of: what an arrogant and idiot
poster you've sent.
> G. Rodrigues- Tekst uit oorspronkelijk bericht niet weergeven -
>
> - Tekst uit oorspronkelijk bericht weergeven -
Gonçalo Rodrigues
I've sent a post, not a poster. Anyway, in one thing you are correct,
I have descended to personal insult and for that my appologies. I will
try to refrain from doing that in the future.
G. Rodrigues
Han de Bruijn
With all the examples mentioned, there is a path of heuristic reasoning
from the seen to the unseen. I've gone such paths while I was a physics
student. The unseen does not just come out of the big blue sky.
>>As a theoretical physicist, I have a decent (non naive) understanding of
>>reality. I know that (actual) infinity is not in there. But I'll respect
>>your choice.
>
> A theoretical physicist?
>
> I have by me the book "Quantum Mechanics", second edition, by E.
> Merzbacher, from which I learned the subject a few years ago (along
> with some others like the "Quantum Mechanics" by L. Landau). Turning
> to chapter 8. entitled "Principles of Wave Mechanics", on the start of
> the second section we can read:
>
> "We have learned that every physical quantity F can be represented by
> a linear operator, which for convenience is denoted by the same
> letter, F."
>
> If we go on we also learn that these linear operators are also
> self-adjoint - so that their spectrum is real. Self-adjointness is
> tied to existence of an inner product in the space. Turning to chapter
> 14 we would learn that the spaces where these operators act are
> Hilbert spaces, which in general are separable but *not necessarily*
> of finite dimension. A separable infinite-dimensional Hilbert space is
> as infinite as infinite gets.
But it can be considered as the limiting case of a finite vector space.
When represented for calculation purposes, you can introduce a cut-off
on space dimension and yet have an arbitrarily accurate approximation.
The fact that Quantum Mechanical vector spaces are all separable, means
that each vector in the infinite space is expressible in the elements of
the basis. Which is the same as for finite vector spaces. You would have
an argument if you could show me a _useful_ infinite space with QM where
the vectors are _not_ expressible in any basis. Because then the latter
fact would be clear evidence that there exist _infinite_ vector spaces
which can _not_ be regarded as a limiting case of finite vector spaces.
With other words: this would prove the applicability of actual infinity
for vector spaces. I'm pretty sure that NO such evidence can be found.
> We could multiply the examples infinitely (pun intended) drawing from
> all sources from Classical Mechanics, Special and General Relativity,
> etc. The fact is, is that all the successful theories that model our
> universe use highly infinitary mathematical tools.
They use moderately infinitary mathematical tools. The infinitary herein
can always be looked upon as "very large but I don't know and don't care
how large" but nevertheless finitary. See the example above.
> But do not let these simple facts spoil your fantasies.
Not fantasies. Knowledge.
Han de Bruijn
Han de Bruijn
HdB does not understand mathematics? Just a _few_ references:
http://hdebruijn.soo.dto.tudelft.nl/jaar2004/purified.pdf
http://www.xs4all.nl/~westy31/Electric.html#Irregular
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf
http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/Splines.pdf
http://hdebruijn.soo.dto.tudelft.nl/www/grondig/crossing.htm
http://hdebruijn.soo.dto.tudelft.nl/jaar2006/kromming.pdf
> Never mind reducing the function of art to the "entertainment" of
> others. This is the typical inanity spewed by a complete inability to
> make fine distinctions, and joining in the same phrase, say, the
> Divine Comedy, and the latest Batman comic strip.
HdB does not understand what art is? Just a _few_ references:
http://hdebruijn.soo.dto.tudelft.nl/www/muziek/download.htm
http://hdebruijn.soo.dto.tudelft.nl/omastijd/index.htm
> By God, what an arrogant idiot you are.
HdB is arrogant? Meaning that he is uncapable of admiring other people:
http://hdebruijn.soo.dto.tudelft.nl/www/muziek/edwin.htm
> G. Rodrigues
Han de Bruijn
Han de Bruijn
Sure. It's just limited to the whole physical universe. If you want to
claim that God's creation is "small" when compared with your virtually
unlimited fantasy, be my guest ..
Han de Bruijn
Han de Bruijn
> IMO, "useful" is the most _useless_ word there is
> when it comes to describing mathematics.
It would be useful to have a mathematical definition of "useful".
I'm not kidding. It could be something along the lines of being
input for functions. The more functions something is input for,
the more useful it is. Just thinking wildly .. Any ideas?
Han de Bruijn
Han de Bruijn
> On 2008-01-09, in sci.math, Han de Bruijn wrote:
>
>>There is a _crucial difference_. Artists, musicians, novelists, actors,
>>screenwriters, professional athletes, chess players, they all entertain
>>_other_ people, rather than themselves.
>
> There are many fields where artists arguably entertain only each other.
So-called Modern Art?
Han de Bruijn
Gonçalo Rodrigues
If I understand you right, you agree with me that in tackling the
difficult problems that theoretical physics poses, *abstract*
mathematical tools were needed time and time again.
So what you are decrying is unmotivated abstraction. I can agree to
that, but the principle is so general that really affords no
refutation, and more importantly, it is useless because it does not
give any objective criteria by which to decide which mathematics is
"good" or "bad". Many fields of mathematics were investigated as pure
mathematics before having any sort of application outside of
mathematics itself.
>>>As a theoretical physicist, I have a decent (non naive) understanding of
>>>reality. I know that (actual) infinity is not in there. But I'll respect
>>>your choice.
>>
>> A theoretical physicist?
>>
>> I have by me the book "Quantum Mechanics", second edition, by E.
>> Merzbacher, from which I learned the subject a few years ago (along
>> with some others like the "Quantum Mechanics" by L. Landau). Turning
>> to chapter 8. entitled "Principles of Wave Mechanics", on the start of
>> the second section we can read:
>>
>> "We have learned that every physical quantity F can be represented by
>> a linear operator, which for convenience is denoted by the same
>> letter, F."
>>
>> If we go on we also learn that these linear operators are also
>> self-adjoint - so that their spectrum is real. Self-adjointness is
>> tied to existence of an inner product in the space. Turning to chapter
>> 14 we would learn that the spaces where these operators act are
>> Hilbert spaces, which in general are separable but *not necessarily*
>> of finite dimension. A separable infinite-dimensional Hilbert space is
>> as infinite as infinite gets.
>
>But it can be considered as the limiting case of a finite vector space.
>When represented for calculation purposes, you can introduce a cut-off
>on space dimension and yet have an arbitrarily accurate approximation.
>
First, you are not really telling me that that is how physicists do
things, are you? Because that is completely false. They simply work in
the infinite dimensional space and have no qualms about it. There is
no intrinsic rule in QM that says that infinite dimensional state
spaces are not possible. In fact, browse through any textbook on QM
and almost all examples are infinite dimensional, from free particles
to the Hydrogen atom. They were there from the start.
Second, no, an infinite dimensional space is not just "the limiting
case of a finite vector space," or more correctly, there is no obvious
unique sense in which it is so. In some cases it is intuitively clear
how this can be done (e.g. the state space for the Hydrogen atom) in
others it is not. And in some cases it is simply *wrong*, in the
specific sense that looking at the action of an operator (= a
measurable quantity in QM) on finite dimensional spaces does *not*
give "an arbitrarily accurate approximation". But all this, it just
says that infinite dimensional state spaces are unavoidable.
>The fact that Quantum Mechanical vector spaces are all separable, means
>that each vector in the infinite space is expressible in the elements of
>the basis. Which is the same as for finite vector spaces. You would have
>an argument if you could show me a _useful_ infinite space with QM where
>the vectors are _not_ expressible in any basis. Because then the latter
>fact would be clear evidence that there exist _infinite_ vector spaces
>which can _not_ be regarded as a limiting case of finite vector spaces.
>With other words: this would prove the applicability of actual infinity
>for vector spaces. I'm pretty sure that NO such evidence can be found.
>
Huh? *Every* Hilbert space has an orthonormal basis (choice needed
here). Separability just means that the orthonormal basis is at most
*countably infinite*. And yes, you need the *full countably infinite*
set of orthonormal vectors to express every vector. So what you are
saying doesn't make a lick of a sense. Perhaps, you are confusing the
notions of Hamel basis and orthonormal basis? In finite dimensional
spaces there are no big differences between the two concepts, but
definitely *not so* in infinite dimensions.
So there it is for your "evidence". To add to the above:
- First, you have no compelling *physical* reason (besides your own
prejudice, that is) to rule out infinite dimensional state spaces.
Even in the cases where you can argue that you can introduce a finite
dimensional cut-off like the Hydrogen atom, you need the infinite
dimensional space to prove certain things that you can compare with
the lab experiments.
- There are many situations where infinite dimensional state spaces
are simply unavoidable, e.g. quantizing classical systems with
infinitely many degrees of freedom like Q.E.D., or simply necessary to
even be able to talk about certain things (many particle systems,
thermodynamic limits, etc.).
- You cannot simply wave your hands and say "oh, you can approximate
by finite dimensional spaces", that is a cop-out answer that tells us
nothing. You have to to say exactly what do you mean by that. In the
above cases, even if it is possible to do so (a big if), it is also a
highly *nontrivial* task to do it. If you can do it, better be
prepared to win the Nobel prize or something.
>> We could multiply the examples infinitely (pun intended) drawing from
>> all sources from Classical Mechanics, Special and General Relativity,
>> etc. The fact is, is that all the successful theories that model our
>> universe use highly infinitary mathematical tools.
>
>They use moderately infinitary mathematical tools. The infinitary herein
>can always be looked upon as "very large but I don't know and don't care
>how large" but nevertheless finitary. See the example above.
>
So your qualm is the difference between the words "moderatly" and
highly"?
To repeat myself: no, not all cases of infinity in physics are of the
"very large but I don't know and don't care how large" type. This is
simply wrong. Second, even if it were, it was incumbent upon *you* to
prove that, since that is the way present-day theoretical physicists
work: they just work directly on infinite dimensional spaces and have
no qualms about it. Third, even if you proved it, why would
theoretical physicists ditch what they already know?
Regards,
G. Rodrigues
Virgil
If your god's universe is as limited as yours, then your "god's
creation" is too small.
>
> Han de Bruijn
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
"Usefulness" is no more relevant to mathematical creativity than it is
to poetic creativity.
"Useful poetry" is that which appears in birthday cards.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
But it is always later generations, often much later, who eventually
determine what is valuable in previous generations' "modern art".
No generation need be a valid judge of its own.
Gonçalo Rodrigues
<Han.de...@DTO.TUDelft.NL> fed this fish to the penguins:
[snip]
>> You yourself do not *understand* mathematics, you do not enjoy it, and
>> from your own personal bias and prejudice, you leap to the conclusion
>> that no one but mathematicians enjoys mathematics. Wow.
>
>HdB does not understand mathematics? Just a _few_ references:
>
>http://hdebruijn.soo.dto.tudelft.nl/jaar2004/purified.pdf
>http://www.xs4all.nl/~westy31/Electric.html#Irregular
>http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf
>http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf
>http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/Splines.pdf
>http://hdebruijn.soo.dto.tudelft.nl/www/grondig/crossing.htm
>http://hdebruijn.soo.dto.tudelft.nl/jaar2006/kromming.pdf
>
The "you do not understand mathematics" part is solely based on your
blunders and misconprehensions on logic, set theory and analysis, even
on the nature of mathematics itself, on several posts of yours in this
newsgroup. Listing some pdf's will not change that.
>> Never mind reducing the function of art to the "entertainment" of
>> others. This is the typical inanity spewed by a complete inability to
>> make fine distinctions, and joining in the same phrase, say, the
>> Divine Comedy, and the latest Batman comic strip.
>
>HdB does not understand what art is? Just a _few_ references:
>
>http://hdebruijn.soo.dto.tudelft.nl/www/muziek/download.htm
>http://hdebruijn.soo.dto.tudelft.nl/omastijd/index.htm
>
I am sorry, but it was you that reduced the function of art to
entertainment. I quote:
"There is a _crucial difference_. Artists, musicians, novelists,
actors, screenwriters, professional athletes, chess players, they all
entertain _other_ people, rather than themselves. While mathematicians
tend to be quite boring for virtually anybody else than themselves."
Besides that last sentence, which is nothing more than gratuitous
insult, if that is not misunderstanding art I don't know what is. If
you meant something else, then you should have qualified your
sentences. I am not a mind reader.
>> By God, what an arrogant idiot you are.
>
>HdB is arrogant? Meaning that he is uncapable of admiring other people:
>
>http://hdebruijn.soo.dto.tudelft.nl/www/muziek/edwin.htm
>
Huh? Do you know what "arrogant" means? I suggest you consult a
dictionary.
Regards,
G. Rodrigues
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> There are many fields where artists arguably entertain only each other.
>
> So-called Modern Art?
Some of it, perhaps, but probably you have in mind rather some of
contemporary art. Modern art, amusingly, is often quite old and canonised by
now.
tommy1729
> <Han.de...@DTO.TUDelft.NL> fed this fish to the
> penguins:
whats with the penguin ?
Gonçalo Rodrigues wrote:
(snip)
Third, even if you proved it, why
> would
> theoretical physicists ditch what they already know?
well we kinda ditched newton and replaced it with relativity ...
of course newton's knowledge is still educated for its simplicity and trivial insights.
but the point is wrong theorems get ditched all the time... because they are wrong of course.
as for current physics i personally wonder if those infinite dimensions are really neccesary.
perhaps 150 dimensions are enough.
and considering equations ( differential equations mainly ) i can tell you ( you already know i assume )
that 3 dimensions already have a lot of dynamics and complexity ( e.g. 3d fractals and navier-stokes eq )
however i admit not being an expert at infinite dimensional vectors or physics , just my gut feeling.
i prefer to support hidden variable theories.
>
> Regards,
> G. Rodrigues
regards
tommy1729
Han de Bruijn
> i prefer to support hidden variable theories.
Why Tommy, why?
Han de Bruijn