Infinity Again
apoorv
(1-1/3), (1-1/4) and so on.
1)Zeno starts walking from the point 2 towards 0.No two particles
occupy the same point;so he must cross them one at a time.Which is the
particle he crosses first?
2)Zeno rolls an identical particle from the point -1 with a velocity 2
units per second towards the right.The collisions are all assumed
perfectly elastic, so that momentum is conserved. At time =2 sec, a
particle crosses the point 1 with the velocity 2 units per seecond.
Which is this particle?
-Apoorv
apoorv
G.E. Ivey
> at 0, (1-1/2),
> (1-1/3), (1-1/4) and so on.
> 1)Zeno starts walking from the point 2 towards 0.No
> two particles
> occupy the same point;so he must cross them one at a
> time.Which is the
> particle he crosses first?
>
> 2)Zeno rolls an identical particle from the point -1
> with a velocity 2
> units per second towards the right.The collisions are
> all assumed
> perfectly elastic, so that momentum is conserved. At
> time =2 sec, a
> particle crosses the point 1 with the velocity 2
> units per seecond.
> Which is this particle?
There is no such particle. If you are going to talk physics, not mathematics, then you cannot place particles wherever you want.
> -Apoorv
>
mathman
2) Physically impossible - mathematically, there is none.
apoorv
G.E. Ivey wrote:
> > Identical Particles are placed on the segment [0,1)
> > at 0, (1-1/2),
> > (1-1/3), (1-1/4) and so on.
> > 1)Zeno starts walking from the point 2 towards 0.No
> > two particles
> > occupy the same point;so he must cross them one at a
> > time.Which is the
> > particle he crosses first?
> >
> There is no "first" particle. What makes you think there should be?
started, he had not crossed a single particle;when he is at 0, he has
crossed all the particles. Surely, there must be an identifiable
instant between these two instants, when he has crossed just one
particle --- which is that instant and which is that particle?
> > 2)Zeno rolls an identical particle from the point -1
> > with a velocity 2
> > units per second towards the right.The collisions are
> > all assumed
> > perfectly elastic, so that momentum is conserved. At
> > time =2 sec, a
> > particle crosses the point 1 with the velocity 2
> > units per seecond.
> > Which is this particle?
> There is no such particle.
> If you are going to talk physics, not mathematics, then you cannot place particles wherever you want.
any point with infinite precision.
> > -Apoorv
> >
David Marcus
> G.E. Ivey wrote:
> > > Identical Particles are placed on the segment [0,1)
> > > at 0, (1-1/2),
> > > (1-1/3), (1-1/4) and so on.
> > > 1)Zeno starts walking from the point 2 towards 0.No
> > > two particles
> > > occupy the same point;so he must cross them one at a
> > > time.Which is the
> > > particle he crosses first?
> > >
> > There is no "first" particle. What makes you think there should be?
> The particles are crossed sequentially, one at a time. When Zeno
> started, he had not crossed a single particle;when he is at 0, he has
> crossed all the particles. Surely, there must be an identifiable
> instant between these two instants, when he has crossed just one
> particle --- which is that instant and which is that particle?
Why do you say "Surely, there must be an identifiable instant between
these two instants, when he has crossed just one particle"? Saying
"surely" doesn't make it so.
You are asking what is the largest number in the set {1 - 1/n | n=
1,2,3,...}. This set doesn't contain a largest number.
--
David Marcus
tom
I think both questions amount to asking 'what is the biggest number?'
This is tricky because there are different degrees of infinity.. so although natural numbers never go higher than the ordinal omega, you could go further depending on how far you mean by 'and so on...'.
See ordinal and cardinal numbers in wikipedia.org.
I have a quick question here too. If omega is the first ordinal after all the natural numbers, is it valid to say that the largest natural number is omega-1?
Jesse F. Hughes
> G.E. Ivey wrote:
>> > Identical Particles are placed on the segment [0,1)
>> > at 0, (1-1/2),
>> > (1-1/3), (1-1/4) and so on.
>> > 1)Zeno starts walking from the point 2 towards 0.No
>> > two particles
>> > occupy the same point;so he must cross them one at a
>> > time.Which is the
>> > particle he crosses first?
>> >
>> There is no "first" particle. What makes you think there should be?
> The particles are crossed sequentially, one at a time. When Zeno
> started, he had not crossed a single particle;when he is at 0, he has
> crossed all the particles. Surely, there must be an identifiable
> instant between these two instants, when he has crossed just one
> particle --- which is that instant and which is that particle?
Sorry, what was the argument that justifies "surely"?
--
"If .999... = 1 then (.999...)/1 should equal 1
let's see
(.999...)/1 = .999...
[Therefore] .999... still=/= 1" -- An astonishing proof by "S. Enterprize"
Tonico
tom ha escrito:
**************************************************
No, it is not. Omega is a limit ordinal, not a consecutive ordinal of
other one.
Regards
Tonio
tom
Thanks
Tom.
Jesse F. Hughes
> I have a quick question here too. If omega is the first ordinal
> after all the natural numbers, is it valid to say that the largest
> natural number is omega-1?
Just as valid as saying that a round triangle is an equatorial polar
bear.
--
"Your people are about denial. Dreams versus reality. TELLING
yourselves you are great. Telling yourselves you are brilliant.
Telling yourselves you understand mathematics."
--James S. Harris: So obvious that it's kind of sad.
Eckard Blumschein
> I think both questions amount to asking 'what is the biggest number?'
> This is tricky because there are different degrees of infinity.
Really?
Do not tell me the whole boring Dedekind/Cantor stuff including
cardinalities. It is too obviously unfounded.
I prefer logically clean reasoning: The property infinite belongs to a
process which is thought to never end, in particular a process of
getting bigger and bigger or smaller and smaller.
Archimedes axiomatically applied this property to numbers. As long as we
regard numbers like an abstract concept that is independent of physical
limitation, there is no biggest number at all and also no smallest p/q.
Aristotele correspondingly concluded that infinity is something that can
never actually be reached. There is no perfect alias actual infinity.
In famous Latin words: Infinitum actu non datur. Finite and infinite
mutually exclude each other.
When Spinoza defined infinity like something that cannot be enlarged, he
referred to the not really executable idea of actual infinity. We may
add: This ideal quality of something thought to be infinitely large,
cannot be exhausted as well.
Infinitum, i.e. the infinite, is primarily the anticipated end of a
never ending process. The belonging irreal result of an anticipated
infinitely lasting counting has been denoted with the symbol oo by John
Wallis (1616-1703). There is no possibility to quantify the symbol oo
like a number. Therefore expressions like oo-oo are pointless. This is
made obvious by stunningly correct relationships like oo+a=oo, where a
is any number, and oo is not a number.
Aristotele distinguished between between potential and actual infinity.
While this distinction is reasonable, it must not be interpreted like an
indication for different degrees of infinity but for two different
aspects at two different levels of abstraction.
The wording "potential infinity" is a bit misleading in so far as one
could imagine the process of concern having the potential alias the
possibility to end somewhere. It doesn't have this potential.
Nonetheless, the notion of potential infinity is a very reasonable one.
When writing x->oo, we are taking the realistic point of view, as if we
were counting a potentially infinite quantity x, number by number.
Calling x potentially infinite, we obey that oo cannot be reached.
Notice, oo always denotes the fiction of actual infinity, while we
attribute to x the property of being potentially infinite.
Georg Cantor was correct when he stessed that the potential infinity is
not the genuine infinity. He did, however, not have any hard evidence
for his intuitively based claim that actually infinite numbers do exist.
He just believed in this idea, and he managed making almost the whole
community of mathematicians to share this at least questionable and
obviously unnecessary belief up to now.
I consider the view of Leibniz more reasonable and overly successful.
Leibniz called the infinite a useful fiction with a fundamentum in re.
Gauss was certainly quite right when he called this fictitious infinity
just a "facon de parler [phrase, E.B.] indem man eigentlich von Grenzen
spricht, denen gewisse Verhaeltnisse so nahe kommen als man will,
waehrend anderen ohne Einschraenkung zu wachsen verstattet ist."
Fraenkel's book "Einleitung in die Mengenlehre" (I refer to the 2nd ed.
1923) claims: "Cantor ... blieb es vorbehalten, die Gauss'sche
Behauptung in dem Sinn, in dem sie verstanden worden ist, nicht nur zu
bekaempfen sondern auch zu widerlegen und dem Begriff des
Unendlichgrossen das Buergerrecht im mathematischen Koenigreich zu
verschaffen". Widerlegt sehe ich Gauss nicht.
As well known to us, there is presumably no alternative but to postulate
the exact numerical representation of any incommensurable ratio to be
hidden within an endless and therefore uncountable amount of numerals.
Incommensurable means irrational. I.e. there is no representation like
p/q with finite p and q.
What about countability: Anything that consists of discrete Elements is
countable. The numbers 1, 2, 3, ... are of course countable, no matter
whether or not this series ends. In principle, this counting can be
performed beyond any limit and deliver a set of countable natural numbers.
However, on a higher level of abstraction there is an alternative
anticipating way of thinking. It restricts to consideration of the
unrealistic entity of all possible natural numbers altogether. This
unresolvable "set" of indefinitely much of natural numbers is definitely
not countable. Because it has to contain indefinitely much of fictious
elements, it looks continuous to everybody who takes it as a whole.
In other words: Being uncountably infinite is not a higher degree of
being infinite but a different quality in addition to it. Therefore one
must not quantitatively compare real "numbers" with rational genuine
numbers. The reals are uncountable. This implies: There are _not_ more
reals than rationals. Something that cannot be counted evades any
numeral comparison. If one respects the continuum of reals to be an
entity being thought of fictitious elements of the same sort altogether,
than it does not contain addressable rationals any more but it
constitutes something unresolvable.
Eckard Blumschein
Eckard Blumschein
> Eckard Blumschein wrote:
>> Fraenkel's book "Einleitung in die Mengenlehre" (I refer to the 2nd ed.
>> 1923) claims: "Cantor ... blieb es vorbehalten, die Gauss'sche
>> Behauptung in dem Sinn, in dem sie verstanden worden ist, nicht nur zu
>> bekaempfen sondern auch zu widerlegen und dem Begriff des
>> Unendlichgrossen das Buergerrecht im mathematischen Koenigreich zu
>> verschaffen". Widerlegt sehe ich Gauss nicht.
I apologize for continuing in German. Last sentence should read: I do
not see Gauss refuted.
> Of all the above I think I'd rather take: "you can't say there are more
> reals than rationals" . Yes, I agree: the adverb "more" here is
> unappliable
I am delighted.
> since we're used to use "more" as comparative of physical,
> finite quantities. Ok.
No no. This comparison does not depend on physics.
> Let's then just say that the rationals Q are embeddable into the reals
> R, and the other way around is NOT possible.
The word "embedded" is somewhat misleading since normally nobody would
expect loosing his identity when embedded let's say into the huge
Chinese population while embedding into the departed more accurately
describes the relationship between irrationals and embedded rationals.
Embedding of the deceased into the living men is indeed impossible.
> Educated mathematicians
> understand from this that this means that the infinity of R is NOT like
> the infinity of Q in THIS respect (the embedding),
Infinity is infinity, as I tried to explain in my preceeding posting.
I consider people miseducated when then believe in different degrees of
infinity. Logic provides a simple, plausible and in the end compelling
explanation for the undoubted fact that the reals are uncountable as is
our notion of a continuum, e.g. like a liquid.
> but if some have hard time swallowing that then let everyone deduce whatever they want
> from this.
Fortunately, I was never urged to swallow the nonsense - or more
political correctly speaking - the highly questionable belief by
Dedekind and Cantor. I had to somehow arrange with political dogma until
1989 being similar in that it also was an Utopia.
The first reason for me to thoroughly deal with the matter were moot
points in connection with the symmetry of R but lacking symmetry of Q.
Once aware of something fundamental being possibly wrong, or at least
unfounded, I came to firm and so far unrefuted conclusion that D&C set
the course into an admittedly benign deadlock.
Kind regards,
Eckard
Tonico
Eckard Blumschein wrote:
> On 11/8/2006 2:23 PM, Tonico wrote:
> > Eckard Blumschein wrote:
>
>
> >> Fraenkel's book "Einleitung in die Mengenlehre" (I refer to the 2nd ed.
> >> 1923) claims: "Cantor ... blieb es vorbehalten, die Gauss'sche
> >> Behauptung in dem Sinn, in dem sie verstanden worden ist, nicht nur zu
> >> bekaempfen sondern auch zu widerlegen und dem Begriff des
> >> Unendlichgrossen das Buergerrecht im mathematischen Koenigreich zu
> >> verschaffen". Widerlegt sehe ich Gauss nicht.
>
> I apologize for continuing in German. Last sentence should read: I do
> not see Gauss refuted.
>
>
> > Of all the above I think I'd rather take: "you can't say there are more
> > reals than rationals" . Yes, I agree: the adverb "more" here is
> > unappliable
>
> I am delighted.
At least you're having a good time. I am delighted (oops! Me too...)
**********************************************************
> > since we're used to use "more" as comparative of physical,
> > finite quantities. Ok.
>
> No no. This comparison does not depend on physics.
That's the way you used. it
***********************************************************
> > Let's then just say that the rationals Q are embeddable into the reals
> > R, and the other way around is NOT possible.
>
> The word "embedded" is somewhat misleading since normally nobody would
> expect loosing his identity when embedded let's say into the huge
> Chinese population while embedding into the departed more accurately
> describes the relationship between irrationals and embedded rationals.
> Embedding of the deceased into the living men is indeed impossible.
*****************************************************************
In mathematics the word "embedded" is perfectly clear: it mean there's
a 1-1 morphism from one object within a category into another. In the
case of groups, this means there's a group monomorphism, in the case of
rings a ring monomorphism...and in the case of sets, that there's a
1-1 function.
As somebody being embedded into the chinese population: the mere
thought of that gives me the chills, and since this is not related to
maths I'd rather won't think of it.
*************************************************************************
>
> > Educated mathematicians
> > understand from this that this means that the infinity of R is NOT like
> > the infinity of Q in THIS respect (the embedding),
>
> Infinity is infinity, as I tried to explain in my preceeding posting.
> I consider people miseducated when then believe in different degrees of
> infinity. Logic provides a simple, plausible and in the end compelling
> explanation for the undoubted fact that the reals are uncountable as is
> our notion of a continuum, e.g. like a liquid.
********************************************************************************
Infinity is infinity as you seem to believe. Ok. I consider poorle
educated (in maths) anyone not knowing that there are 1-1 embeddings
from infinite sets into other infinite sets, without the possibility of
reversing the direction of the embeddings. Call this different degrees
of infinities, different cardinales or however you like. There's no
logic whatsoever that I'm aware of (my bad, probabily) that provides
explanation to the fact that the real numbers set is uncountable other
than the proof of it, either by Cantor's diagonalization Method or by
any other method.
**********************************************************************************
> > but if some have hard time swallowing that then let everyone deduce whatever they want
> > from this.
>
> Fortunately, I was never urged to swallow the nonsense - or more
> political correctly speaking - the highly questionable belief by
> Dedekind and Cantor. I had to somehow arrange with political dogma until
> 1989 being similar in that it also was an Utopia.
****************************************************************
I don't know what belief of Cantor and Dedekind are you talking about,
though I can guess...:)
About you arranging (what?) with political dogma...oh well, I am sorry
you did.
Regards
Tonio
******************************************************************
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> Infinity is infinity, as I tried to explain in my preceeding posting.
> I consider people miseducated when then believe in different degrees of
> infinity.
And we have as least as good reason to consider those who cannot
distinguish different qualities of infiniteness both stupid and
ignorant.
> Logic provides a simple, plausible and in the end compelling
> explanation for the undoubted fact that the reals are uncountable as is
> our notion of a continuum, e.g. like a liquid.
Logic does not explain how to pour out a glass full of real numbers, nor
does it justify any such physical analogy to something that is
physically impossible.
>
> > but if some have hard time swallowing that then let everyone deduce
> > whatever they want
> > from this.
>
> Fortunately, I was never urged to swallow the nonsense
EB does it voluntarily, without any urging.
> political correctly speaking
Your speaking is no more politically correct than it is mathematically
correct. It is not correct at all.
>
> The first reason for me to thoroughly deal with the matter were moot
> points in connection with the symmetry of R but lacking symmetry of Q.
There is no symmetry of R in any numerical sense that Q lacks.
> Once aware of something fundamental being possibly wrong, or at least
> unfounded, I came to firm and so far unrefuted conclusion that D&C set
> the course into an admittedly benign deadlock.
EB certainly seems to have deadlocked his thought processes into a
useless tangle, but his problems do not affect standard mathematics at
all.
Eckard Blumschein
> Eckard Blumschein wrote:
>> On 11/8/2006 2:23 PM, Tonico wrote:
>> > Eckard Blumschein wrote:
>> > since we're used to use "more" as comparative of physical,
>> > finite quantities. Ok.
>>
>> No no. This comparison does not depend on physics.
> **********************************************************
> That's the way you used. it
> ***********************************************************
Do you have any indication for that suspicion?
More, equal and less always relate to counting, i.e. to countable
numbers only.
According to even Fraenkel, the fourth logical possibility is "uncomparable"
How can there be a fourth possibility outside counting?
The answer is quite simple: Fraenkel effectively did not at all deal
with counting and genuine numbers but with the terms smaller, equally
large and larger instead. While this classification is also independent
of its physical interpretation, it is wider. It includes continuous
measures too. Therefore it fits to geometry. Perhaps, some narrow-minded
mathematicians already confuse geometry with physics.
We may benefit from shedding light into basic confusion by Dedekind.
Dedekind defined infinity by using the notion of equivalence.
Equivalence belongs to the four possibilities for the extensional
measure a and b:
a is smaller than b (b includes a, but a does not include b)
a equals b (b includes a, and a includes b)
a is larger than b (a includes b, but b does not include a)
a is uncomparable with b (a and b do not overlap)
Once again: For numbers, there are only three possibilities: Trichotomy.
Tertium non datur (TND) means: There is no third possiblilty. A number
has to either exist or not.
Intuitionists like Brouwer found out: TND fails with perfect infinity.
Vaughan S. Pratt explained: The continuum is the third.
Now we can get aware that the notion of an infinite set tries to make
possible the impossible: Making countable the uncountable. This only
works inside brain/theory and only with logical split. Notice: I do not
suspect Cantor's mental illness schizophrenia. On the contrary. Cantor
might have been a mentally sound person under enormous pressure due to
not entirely overcome feeling to be possibly wrong. No no, I refer to
the basic logical contradition. A set of single elements can not be
continuous and simultaneously maintain countability of its elements.
While rational numbers are countable and belong to the world of numbers
being characterized by trichotomy, real "numbers" are uncountable due to
lacking trichotomy and belong to the realm of continuum.
>
>> > Let's then just say that the rationals Q are embeddable into the reals
>> > R, and the other way around is NOT possible.
>>
>> The word "embedded" is somewhat misleading since normally nobody would
>> expect loosing his identity when embedded let's say into the huge
>> Chinese population while embedding into the departed more accurately
>> describes the relationship between irrationals and embedded rationals.
>> Embedding of the deceased into the living men is indeed impossible.
> *****************************************************************
> In mathematics the word "embedded" is perfectly clear:
As long as "mathematics" is not yet expelled from Cantor's paradise.
Let's look whether it is worth to maintain pertaining notions:
it mean there's
> a 1-1 morphism from one object within a category into another.
Maybe the word category has meanwhile a meaning that is quite deviating
from traditional science outside modern definomania. Hopefully you will
understand me though. Isn't a countable rational number a quite
different element being not at all comparable with any uncountable
because fictitious real "number"?
> As somebody being embedded into the chinese population: the mere
> thought of that gives me the chills, and since this is not related to
> maths I'd rather won't think of it.
> *************************************************************************
Do you not feel a bit like a racist? Indeed has to distinct between
rationals and reals much more categorically. Black and white people may
have very nice children together. The categorical distinctness of real
"numbers" cannot be bridged. Even those mathematicians who are unable to
argue without CF may obey aleph_0 different from aleph_1 with nothing in
between.
Logically: Something can be either countable or uncountable, alife or
dead, finite or infinite.
>>
>> > Educated mathematicians
>> > understand from this that this means that the infinity of R is NOT like
>> > the infinity of Q in THIS respect (the embedding),
>>
>> Infinity is infinity, as I tried to explain in my preceeding posting.
>> I consider people miseducated when then believe in different degrees of
>> infinity. Logic provides a simple, plausible and in the end compelling
>> explanation for the undoubted fact that the reals are uncountable as is
>> our notion of a continuum, e.g. like a liquid.
> ********************************************************************************
> Infinity is infinity as you seem to believe. Ok. I consider poorle
> educated (in maths) anyone not knowing
What ignorance to suspect someone who does not share Chritian belief
just needs read the bible. If there were solid evidence in support of
the ideas by Dedekind and Cantor, I would accept it. So far I am seeing
just an widespread illusion, an Utopia.
that there are 1-1 embeddings
> from infinite sets into other infinite sets, without the possibility of
> reversing the direction of the embeddings. Call this different degrees
> of infinities, different cardinales or however you like.
I would never lower my standard.
> There's no
> logic whatsoever that I'm aware of (my bad, probabily) that provides
> explanation to the fact that the real numbers set is uncountable
Really? My logic is quite simple here, and it follows Galilei who
already understood that a quantity is not doomed to be either smaller od
equally large of larger than a second one but may be just uncomparable.
Something that cannot be counted cannot be compared on a numerical basis
and vice versa. Already ancient mathematicians were aware of alogos.
What about your term "real numbers set" I would like to stress that
already the word set is mistakable here. The "set" of real numbers is
just a ficitious one. The also fictitious "real" numbers are anything
else but real, real "numbers" are no numbers in the sense they are not
numerically arressable and therefore they do not obey trichotomy.
> other than the proof of it, either by Cantor's diagonalization Method or by
> any other method.
Cantor's 2nd DA war a wrong proof in sofar as it correctly illustrates
that real numbers are uncountable but the intended interpretation was
definitely wrong.
What about the Power set of IN, I gave an explanation elsewhere.
Can you please point to further evidence?
> **********************************************************************************
>> > but if some have hard time swallowing that then let everyone deduce whatever they want
>> > from this.
>>
>> Fortunately, I was never urged to swallow the nonsense - or more
>> political correctly speaking - the highly questionable belief by
>> Dedekind and Cantor. I had to somehow arrange with political dogma until
>> 1989 being similar in that it also was an Utopia.
> ****************************************************************
> I don't know what belief of Cantor and Dedekind are you talking about,
> though I can guess...:)
> About you arranging (what?) with political dogma...oh well, I am sorry
> you did.
Do you deliberately not understand my words or is my English too poor?
No, I did not refer to religion or politics when I wrote of the belief
_by_ not of Dedekind and Cantor. "By" means not just shared but created
by the two.
> Regards
> Tonio
> ******************************************************************
>
>> The first reason for me to thoroughly deal with the matter were moot
>> points in connection with the symmetry of R but lacking symmetry of Q.
>> Once aware of something fundamental being possibly wrong, or at least
>> unfounded, I came to the firm and so far unrefuted conclusion that D&C set
Eckard Blumschein
> In article <4552E1E7...@et.uni-magdeburg.de>,
> Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
>
>> Infinity is infinity, as I tried to explain in my preceeding posting.
>> I consider people miseducated when then believe in different degrees of
>> infinity.
>
> And we have as least as good reason to consider those who cannot
> distinguish different qualities of infiniteness both stupid and
> ignorant.
I learned to cope with those who were reproaching to me I did not
understand the socialism. My parents had to cope with even more
presumptious and dangerous ideology.
Please do not get me wrong. I do not intend escalating our discussion
towards mutual insult. If you realy have good reasons, you will be in
position to argue factually. However, I rather suspect you were urged to
learn allegedly counterintuitive stuff, and you were told those are jst
dull who cannot grasp it.
>> Logic provides a simple, plausible and in the end compelling
>> explanation for the undoubted fact that the reals are uncountable as is
>> our notion of a continuum, e.g. like a liquid.
>
> Logic does not explain how to pour out a glass full of real numbers, nor
> does it justify any such physical analogy to something that is
> physically impossible.
I spoke a logic explanation and I added just an illustration to an
undoubted fact. The notion of a continuum is an abstract one:
Peirce: Something every part of which has parts.
In order to facilitate your understanding, Stifel (at the time of Martin
Luther), Weyl (before I was born) and I looked for something more
obvious and tangible. Strictly speaking there is no continuum in physics
at all. Physics only benefits from using the ideas numbe and continuum.
>>
>> > but if some have hard time swallowing that then let everyone deduce
>> > whatever they want
>> > from this.
>>
>> Fortunately, I was never urged to swallow the nonsense
>
> EB does it voluntarily, without any urging.
The mathematics I learned was oriented towards powerful application
without unnecessary and rather questionable marginals.
>> political correctly speaking
> Your speaking is no more politically correct than it is mathematically
> correct. It is not correct at all.
Perhaps English is not your mother's tongue.
>>
>> The first reason for me to thoroughly deal with the matter were moot
>> points in connection with the symmetry of R but lacking symmetry of Q.
>
> There is no symmetry of R in any numerical sense that Q lacks.
Scissors can cut a sheet of paper exactly in the middle. Mathematicians
just not yet understand that they are not wrong if they do the same for R.
In Q one has to decide how to deal with zero. I recommend declaring the
nil negative. Elsewhere I explained why.
Tonico
Eckard Blumschein ha escrito:
> On 11/9/2006 10:39 AM, Tonico wrote:
> > Eckard Blumschein wrote:
> >> On 11/8/2006 2:23 PM, Tonico wrote:
.........................................................
> that there are 1-1 embeddings
> > from infinite sets into other infinite sets, without the possibility of
> > reversing the direction of the embeddings. Call this different degrees
> > of infinities, different cardinales or however you like.
> I would never lower my standard.
********************************************************************************
Yes, some standards just can't be lowered...
And yes: your english sometimes is very hard to understand, at least
for me. Anyway, it seems like you're very fond of quoting ancient great
men, like Aristotle say (who by the way was wrong in many, perhaps
most, of the scientific stuff he messed with). I don't know, perhaps
this gives you some sense of security in your opinions.
You also seem to rejoice in using terms like "fictitious
numbers"....fictitious as compared to what? The irrational 2^(1/2) is
more fictitious than the numbers 3, 6 or -2? Why do you think this is
so? What is your definition of "fictitious"? So far these, and other,
numbers have proved extraordinarily useful both within the mathematical
world and in many applications out of it...
You keep on trying, rather pathetically imo, to knock Dedekind and
Cantor's ideas...good luck with that, but I don't think you'll succeed
in doing any harm to a well established, strong theory just by whinning
about it.
Your sense of "logic" doesn't seem to work, at least if we're to judge
from the success your ideas have. These ideas of yours are rather
pretty baseless and seem to point towards serious lacks of good
mathematical education. Your calling "uneducated" to mathematicians
only because they accept a sound theory that you, and others, are
simply unable to understand is pretty comical, but hey!: As long as you
are delighted ....
Dedekindian regards
Tonio
David Marcus
> Isn't a countable rational number a quite
> different element being not at all comparable with any uncountable
> because fictitious real "number"?
Try using words with their normal meanings. It is nonsense to say
"countable rational number" (assuming standard meanings of the words).
And, "fictitious" has no standard meaning, so you must define it, if you
wish to use it.
> Do you deliberately not understand my words or is my English too poor?
You are incoherent. If you insist on using standard words with your own
private meanings, communication is (of course) impossible.
--
David Marcus
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/9/2006 10:39 AM, Tonico wrote:
> > Eckard Blumschein wrote:
> >> On 11/8/2006 2:23 PM, Tonico wrote:
> >> > Eckard Blumschein wrote:
>
> >> > since we're used to use "more" as comparative of physical,
> >> > finite quantities. Ok.
> >>
> >> No no. This comparison does not depend on physics.
> > **********************************************************
> > That's the way you used. it
> > ***********************************************************
>
> Do you have any indication for that suspicion?
> More, equal and less always relate to counting, i.e. to countable
> numbers only.
More, less or equal are also used in physical comparisons which are only
incidentally numerical. To say that one person is taller (or stronger
or faster or heavier or smarter) than another does not require any
numbers.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
Which abstraction can easily be avoided by considering the ordinal
property of completness in its stead.
>
> >> political correctly speaking
> > Your speaking is no more politically correct than it is mathematically
> > correct. It is not correct at all.
>
> Perhaps English is not your mother's tongue.
It was my mother's mother tongue as it is mine.
>
>
> >>
> >> The first reason for me to thoroughly deal with the matter were moot
> >> points in connection with the symmetry of R but lacking symmetry of Q.
> >
> > There is no symmetry of R in any numerical sense that Q lacks.
>
> Scissors can cut a sheet of paper exactly in the middle. Mathematicians
> just not yet understand that they are not wrong if they do the same for R.
I very much doubt that any piece of paper can be cut so exactly in half
that no difference between the pieces can be detected.
And the real numbers cannot be cut that way either without the cut point
being asymmetrically belonging to one side or the other but not both.
Now the rationals CAN be cut into ordinally symmetric mirror image
halves by requiring the LUB of the lower half and GLB to the upper not
to be any rational number.
Ross A. Finlayson
> >
> > And we have as least as good reason to consider those who cannot
> > distinguish different qualities of infiniteness both stupid and
> > ignorant.
>
Half of the integers are even.
A proper superset is larger than the regular set.
The reals are obviously "larger" than the integers, rationals, or
irrationals, where each of those are proper subsets of the reals.
So, Virgil, are you stupid, or ignorant? No: both.
Virgil, whoever your "we" is: no, you don't.
Ross
Tonico
Ross A. Finlayson ha escrito:
> > On 11/9/2006 11:15 AM, Virgil wrote:
> > >
> > > And we have as least as good reason to consider those who cannot
> > > distinguish different qualities of infiniteness both stupid and
> > > ignorant.
> >
>
> Half of the integers are even.
>
> A proper superset is larger than the regular set.
************************************************************************
Unless you define what you mean by "larger" this is nonsense: the
naturals are a subset of the integers...are the integers "Larger" than
the naturals?? They both are equipotent. What do you mean? Common, be
like-a-mathematician and define your stuff.
The same applies to the following stuff.
Regards
Tonio
***************************************************************************
Ross A. Finlayson
Hi Toni,
I put forth a wide variety of arguments here to that effect.
People said I was wrong.
Fred Katz wrote a Ph.D. thesis to the effect of that a proper subset is
smaller than the set, quite some time ago before I had heard of it.
So, that is not a contentious point anymore.
I was right.
Sets of numbers are sets of numbers and everything about them applies.
Ross
Tonico
Ross A. Finlayson כתב:
Hi Ross:
I don't have the faintest idea who Fred Kats is/was and what he wrote,
but if he wrote a mathematical paper/thesis "to the effect" (does this
mean "proved" or "showed" or something? Or perhaps he just TRIED
something??) that a proper subset is "smaller" than a set that contains
is then I'm sure he MUST have given a comprehensive definition of
"smaller", otherwise his thesis would have been rejected even by the
congress, leave alone by intelligent, academic professors.
Again, I ask: what is the definition of "smaller" or larger" WITH
RESPECT to infinite sets?
Regards
Tonio
Ps. BTW, what did people tell you you were wrong about, and how did you
come to the conclusion they all (who??) were wrong and you were right?
And how does the existence of some mathematical thesis, even a VERY
GOOD ONE, decide that some contentious point (which one) is over? Was
that thesis ever peer-reviewed somewhere? I mean, if there was some
debate going on and someone wrote a thesis and settled the question
then I bet some teams of mathematicians went thru that thesis and
decided it was correct and thus the point was settled. Is this the
case? Again, what the point is?
Thanx
Eckard Blumschein
> Eckard Blumschein ha escrito:
> And yes: your english sometimes is very hard to understand, at least
> for me.
Sorry for my shaky command of English.
> Anyway, it seems like you're very fond of quoting ancient great
> men,
Cantor himself made this necessary because he claimed that they were
wrong while Cantor himself had no tangible evidence for this.
> like Aristotle say (who by the way was wrong in many, perhaps
> most, of the scientific stuff he messed with).
Did you deal with Aristotele?
> You also seem to rejoice in using terms like "fictitious
> numbers"....fictitious as compared to what? The irrational 2^(1/2) is
> more fictitious than the numbers 3, 6 or -2?
There are systems of rational numbers with particular basis, e.g ten.
One can also linearly address any number among all rational numbers.
These numbers have a representation by just potentially infinitely many
numerals. Accordingly, they are countable.
Numerical representations of irrational "numbers" are strictly speaking
no numbers at all. They may be "thought" of like having an actually
infinite, uncountable amount of numbers, in other, more realistic words:
The do not at all have a numerical representation.
So the rationals are countable genuine numbers, while the irrationals
are fictitious uncountable ones?
Yes. This distinction is important.
However, this is not yet the whole story. You know, both irrationals and
embedded rationals together are called real numbers. Do we confuse
genuine and fictitious numbers in this case?
No. Embedded rationals behave as fictitious as irrationals. They lost
their numerical identity.
> Why do you think this is
> so? What is your definition of "fictitious"? So far these, and other,
> numbers have proved extraordinarily useful both within the mathematical
> world and in many applications out of it...
They were useful long before Dedekind and Cantor declared them like
numbers with full civil right.
> You keep on trying, rather pathetically imo, to knock Dedekind and
> Cantor's ideas...good luck with that, but I don't think you'll succeed
> in doing any harm to a well established, strong theory just by whinning
> about it.
I was born into well established, meanwhile almost forgotten isms.
> Your sense of "logic" doesn't seem to work, at least if we're to judge
> from the success your ideas have.
Breakdown of illusions takes time and requires effort.
> Dedekindian regards
Why so childish?
Tonico
Eckard Blumschein ha escrito:
> > > Why do you think this is
> > so? What is your definition of "fictitious"? So far these, and other,
> > numbers have proved extraordinarily useful both within the mathematical
> > world and in many applications out of it...
>
> They were useful long before Dedekind and Cantor declared them like
> numbers with full civil right.
>
> Dedekindian regards
>
> Why so childish?
***************************************************
You talk about Dedekind and Cantor declaring (when? where? how? Why do
YOU think this is true??) numbers with "full civil rights"...and I am
the childish one?? Aha, I see....
DedekindCantorian regards
Tonio
Ps. Not trying to bother, but would you mind to tell what's your
mathematical background? Where and what did you study, etc. Not that
this gives you more or less right to have and express your opiniion,
but it surely can affect the pondering of it, imo.
David R Tribble
>> And we have as least as good reason to consider those who cannot
>> distinguish different qualities of infiniteness both stupid and
>> ignorant.
>
Ross A. Finlayson wrote:
> Half of the integers are even.
True.
> A proper superset is larger than the regular set.
False. The integers are a "proper" superset of the even
integers, but it is not a larger set. If you meant "powerset",
then yes, that's true.
> The reals are obviously "larger" than the integers, rationals, or
> irrationals, where each of those are proper subsets of the reals.
The reals are a larger set than the integers and rationals, but
not larger than the irrationals.
> So, Virgil, are you stupid, or ignorant? No: both.
> Virgil, whoever your "we" is: no, you don't.
You don't appear to know the difference between "countable"
and "uncountable", and you frequently stick an "all infinities are
equivalent" (or words to that effect) into your posts. So his
barb would seem to apply to you.
I withhold judgement, since I usually can't make sense of most
of your longer posts, so I really can't say what your beliefs are.
Eckard Blumschein
> You talk about Dedekind and Cantor declaring (when? where? how? Why do
> YOU think this is true??) numbers with "full civil rights"...and I am
> the childish one??
I have at hands Fraenkel: Einleitung in die Mengenlehre, 2nd ed. 1923,
Perron: Die Irrationalzahlen 1921, Betsch: Fiktionen in der Mathematik, 1926
E. B.
Eckard Blumschein
> The reals are a larger set than the integers and rationals, but
> not larger than the irrationals.
Just because you are sharing this opinion with many I take the
opportunity to object.
Hopefully are you not too old and too hoity-toity.
Since I am not aware of any possibility to count the reals/irrationals,
I consider any comparison with the also infinite rationals unfounded.
Following Galilei, I say:
There are not more reals than rationals.
Likewise there are not more natural numbers than positive natural numbers.
Regards,
Eckard Blumschein
Tonico
And I have in my hands Weber's Lehrbuch der Algebra, Silverman's
Elliptic Functions and ten fingers...so what, again? Lemme guess: since
you have no mathematical education at all, and since you have decided
to rely completely or almost on what past (and sometimes VERy past)
great men said/thought/gave their opinion about in mathematics, this
time is
the same...? Am I right? Ts,ts,ts...ok, as you wish.
Tonio
Eckard Blumschein
> Eckard Blumschein wrote:
>> On 11/13/2006 3:27 PM, Tonico wrote:
>>
>> > You talk about Dedekind and Cantor declaring (when? where? how? Why do
>> > YOU think this is true??) numbers with "full civil rights"...and I am
>> > the childish one??
>>
>> I have at hands Fraenkel: Einleitung in die Mengenlehre, 2nd ed. 1923,
>> Perron: Die Irrationalzahlen 1921, Betsch: Fiktionen in der Mathematik, 1926
>>
>> E. B.
> *************************************************************
> And I have in my hands Weber's Lehrbuch der Algebra, Silverman's
> Elliptic Functions and ten fingers...so what, again?
Didn't you doubt that the expression "full civil rights" is not my
invention? I gave you some sources. Do you need the pages too?
Ross A. Finlayson
Yes, the (asymptotic) density of the even integers in the integers is
one half. The density of the integers within themselves is one. So,
which set is smaller in the integers or any superset of the integers,
where .5 < 1? Is .5 as a magnitude smaller than 1 as a magnitude?
So, in comparing those two sets' (of numbers with well-known
properties) densities as magnitudes, is there not a meaningful
difference? Here I'm not going to bother opening the can of worms
about uniform distributions over the naturals, although a variety of
characteristics could be described for any thing that could be that
thing.
Consider a variety of other well-known asymptotic densities of various
types of numbers within the integers or natural integers. Similarly to
the notion of the infinite sum and whether it exists and is equal to
the limit, as for no finite integer index is it the sum, there are
non-zero and variously rational or irrational known quantities of those
densities, or for example probabilities of pairs of integers at uniform
random being coprime, given in terms of their distribution. Those
asymptotic values only hold for the entirety of the integers.
About a proper subset being in a sense less in some sense of size than
a superset, or as well in another sense a set with infinitely many
proper supersets being proper subsets of a more strongly larger set, eg
in comparing the integers to rationals, there are obviously a wide
variety of ways to qualitatively compare infinite sets of numbers, or
other items. In terms of some eventual ultimate space of numbers, the
big continuum, that's a universal number space, that's right number
space, of sorts, and to be plain it is not addressed by standard
theories.
So, Dave, Dave, Dave, Steve, I was pointing out these things because
Virgil was insulting you, and himself, and everybody else.
Ross
Jan Burse
David R Tribble wrote:
>>Half of the integers are even.
> True.
No.
Here is my proof, 1/3 of the integers are even.
Take the sequence:
x=1 3 2 5 7 4 9 11 6 etc..
Now cn = number of xi i<n and xi even.
Lim n->oo cn/n = 1/3
Proof:
c1=0
c2=0
c3=0.33...
c4=0.25
c5=0.2
c6=0.33...
c3*k+1=k/(3*k+1)
c3*k+2=k/(3*k+2)
c3*k+3=(k+1)(3*k+3)
Lim k->oo k/(3*k+1) = 1/3
Lim k->oo k/(3*k+2) = 1/3
Lim k->oo (k+1)(3*k+3) = 1/3
So there is a single "häuffungspunkt" which is 1/3.
Bye
P.S.: You can construct any real number
via counting evens...
Tonico
Eckard Blumschein ha escrito:
************************************************************
What in the world gives you the idea that I doubted that the expression
isn´t your invention? It just is as childish an expression as any of
mine, or many others. Since I don´t have the book by Fraenkel it´d be
worthless to tell me the pages.
Anyway, it is interesting to note that you did NOT address any of my
really direct and serious objections to your rather unbased and
senseless criticism...ok.
Tonio
Jan Burse
The understanding of r relates to the infinity of the natural numbers
As cheese relates to the moon.
Neither is the moon made of cheese nor
does r understand anything about the
infinity of the natural numbers.
Or maybe I am wrong: Please zoom in here:
http://moon.google.com/
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/14/2006 1:34 AM, David R Tribble wrote:
>
> > The reals are a larger set than the integers and rationals, but
> > not larger than the irrationals.
>
> Just because you are sharing this opinion with many I take the
> opportunity to object.
>
> Hopefully are you not too old and too hoity-toity.
>
> Since I am not aware of any possibility to count the reals/irrationals,
> I consider any comparison with the also infinite rationals unfounded.
> Following Galilei, I say:
> There are not more reals than rationals.
When and where did Galileo Galilei lead that justifies anyone ever to
say anything like that?
> Likewise there are not more natural numbers than positive natural numbers.
That last is quite valid, surprisingly.
>
> Regards,
> Eckard Blumschein
Eckard Blumschein
> mathematical background?
In the next step you could ask for my family, race, belief, etc.
All this will not help against compelling factual arguments of mine.
Well, I am proud of my teacher of mathematics Prof. Nikolaus Joachim
Lehmann who was simultaneously a German pioneer of "small" computers
when I studied Electrical Engineering at University of Dresden from 1960
to 1966. I will never forget him climbing from his blackboard beneath of
the auditory, almost jumping from desk to desk until he reached the
entrance on top. The auditory was overcrowded because Nabla J., as we
called him, had overrun his time limit. We were still sitting behind our
desks while the next students pushed into the auditory from above. Two
of them insulted him. Imagine, he stood beneath at the blackboard. They
felt safe at the pretty distant entrance above. No matter for a
personality like Nabla J, no matter that he was not slim. The two
students could not escape, since there was a bulk of pushing students
behind them. Nabla J. captured them, climbing over the desk we were
sitting behind.
I do not have any problem with my nice family ...
There is nothing that could nurture any hope to disqualify me
even if my last paper does not have much to do with mathematics:
http://iesk.et.uni-magdeburg.de/~blumsche/M283.html
Eckard Blumschein
> Eckard Blumschein ha escrito:
>> Didn't you doubt that the expression "full civil rights" is not my
>> invention? I gave you some sources. Do you need the pages too?
> ************************************************************
> What in the world gives you the idea that I doubted that the expression
> isn´t your invention? It just is as childish
I do not see any reason why (perhaps Cantor himself but at least)
Fraenkel, Perron, and Betsch used a childish expression.
>an expression as any of
> mine, or many others. Since I don´t have the book by Fraenkel it´d be
> worthless to tell me the pages.
It is available in good libraries.
> Anyway, it is interesting to note that you did NOT address any of my
> really direct and serious objections to your rather unbased and
> senseless criticism...ok.
Do not tell lies.
If you maintain that I did not or not clearly enough refute a factual
argument, then please repeate it. Maybe, I did just not have enough tim
to read it and reply to it.
Eckard Blumschein
> In article <4559BAF3...@et.uni-magdeburg.de>,
> Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
>
>> On 11/14/2006 1:34 AM, David R Tribble wrote:
>>
>> > The reals are a larger set than the integers and rationals, but
>> > not larger than the irrationals.
>>
>> Just because you are sharing this opinion with many I take the
>> opportunity to object.
>>
>> Hopefully are you not too old and too hoity-toity.
>>
>> Since I am not aware of any possibility to count the reals/irrationals,
>> I consider any comparison with the also infinite rationals unfounded.
>
>> Following Galilei, I say:
>> There are not more reals than rationals.
>
> When and where did Galileo Galilei lead that justifies anyone ever to
> say anything like that?
Galileo Galilei was arrested. Certainly you know why. So he had time to
thoroghly think.
>
>> Likewise there are not more natural numbers than positive natural numbers.
>
> That last is quite valid, surprisingly.
Cantor was also surprized. Most likely he also did not know, and perhaps
he too would not have understood the logic by Galileo Galilei. He wrote
in French: Je le vois mais je ne le crois pas.
You might find the exact wording of Galilei via pages by Rudolf Sponsel
also who quoted a lot of related stuff.
Tonico
Eckard Blumschein ha escrito:
> On 11/13/2006 3:27 PM, Tonico wrote:
> > would you mind to tell what's your
> > mathematical background?
>
> In the next step you could ask for my family, race, belief, etc.
***********************************************************************
Your race?? Unless you mean 100 meters or Indi 500, why would I give
half a damn about your race when TRYING (pretty naive from me, I know)
to have an actual mathematical debate with you? About your belief I
already know something: you believe you have some reasonable argument
against Set Theory as it is now. Well, you don't, but hey: it's your
belief and you're completely free and entitled to have it.
************************************************************************
> All this will not help against compelling factual arguments of mine.
************************************************************************
Wishful thinking (figure of speech)
************************************************************************\
> Well, I am proud of my teacher of mathematics Prof. Nikolaus Joachim
> Lehmann who was simultaneously a German pioneer of "small" computers
> when I studied Electrical Engineering at University of Dresden from 1960
> to 1966. I will never forget him climbing from his blackboard beneath of
> the auditory, almost jumping from desk to desk until he reached the
> entrance on top. The auditory was overcrowded because Nabla J., as we
> called him, had overrun his time limit. We were still sitting behind our
> desks while the next students pushed into the auditory from above. Two
> of them insulted him. Imagine, he stood beneath at the blackboard. They
> felt safe at the pretty distant entrance above. No matter for a
> personality like Nabla J, no matter that he was not slim. The two
> students could not escape, since there was a bulk of pushing students
> behind them. Nabla J. captured them, climbing over the desk we were
> sitting behind.
*****************************************************************************
Great athlete, apparently, that J. Lehmann professor. As mathematician
I can't say anything: I just don't know him
*****************************************************************************
> I do not have any problem with my nice family ...
*****************************************************************************
That, I'm sure, must be very good for you. I don't care, and I wonder
whether you believe (again a belief) that I said anything implying your
family...
****************************************************************************
> There is nothing that could nurture any hope to disqualify me
> even if my last paper does not have much to do with mathematics:
****************************************************************************
Now in this you're mistaken, imo: your baseless, non-mathematical
like-arguments disquilfy you, and this is reinforced by your lack of
education in maths, whcih I believe is a fair question when trying to
find out who your debater is in a MATHEMATICAL debate. Now I know.
Thanx
Regards
Tonio
> http://iesk.et.uni-magdeburg.de/~blumsche/M283.html
David R Tribble
>> The reals are a larger set than the integers and rationals, but
>> not larger than the irrationals.
>
Eckard Blumschein wrote:
> Just because you are sharing this opinion with many I take the
> opportunity to object.
>
> Hopefully are you not too old and too hoity-toity.
>
> Since I am not aware of any possibility to count the reals/irrationals,
> I consider any comparison with the also infinite rationals unfounded.
> Following Galilei, I say:
> There are not more reals than rationals.
I assume you have some logical argument for this, something that
Galileo would approve of?
We can count all the rationals, using a rather nifty bijection I found
some weeks ago:
S(0) = 0
S(1) = 1
S(2n) = S(n)+1 for n>0
S(2n+1) = 1/S(n)
This sequence produces every positive rational, providing a
one-to-one mapping between the naturals and the positive rationals.
The proof of your claim above will show, what, that the rationals are
actually uncountable?
> Likewise there are not more natural numbers than positive natural numbers.
Of course not, since all naturals are positive numbers.
But there are not more integers than positive integers, either.
David R Tribble
>> And we have as least as good reason to consider those who cannot
>> distinguish different qualities of infiniteness both stupid and
>> ignorant.
>
Ross A. Finlayson wrote:
>> A proper superset is larger than the regular set.
>
David R Tribble wrote:
>> False. The integers are a "proper" superset of the even
>> integers, but it is not a larger set. If you meant "powerset",
>> then yes, that's true.
>
Ross A. Finlayson wrote:
> Yes, the (asymptotic) density of the even integers in the integers is
> one half. The density of the integers within themselves is one. So,
> which set is smaller in the integers or any superset of the integers,
> where .5 < 1? Is .5 as a magnitude smaller than 1 as a magnitude?
>
> So, in comparing those two sets' (of numbers with well-known
> properties) densities as magnitudes, is there not a meaningful
> difference? Here I'm not going to bother opening the can of worms
> about uniform distributions over the naturals, although a variety of
> characteristics could be described for any thing that could be that
> thing.
So the density of the even integers within the integers is 1/2, while
the density of the integers within themselves is (obviously) 1.
Thus the densities of different subsets are different. How is this
"meaningful difference" relevant to the size of the sets?
Ross A. Finlayson wrote:
> Consider a variety of other well-known asymptotic densities of various
> types of numbers within the integers or natural integers. Similarly to
> the notion of the infinite sum and whether it exists and is equal to
> the limit, as for no finite integer index is it the sum, there are
> non-zero and variously rational or irrational known quantities of those
> densities, or for example probabilities of pairs of integers at uniform
> random being coprime, given in terms of their distribution. Those
> asymptotic values only hold for the entirety of the integers.
As usual, I can't make sense of this. Does any of this deal with
relative set sizes?
> About a proper subset being in a sense less in some sense of size than
> a superset, or as well in another sense a set with infinitely many
> proper supersets being proper subsets of a more strongly larger set, eg
> in comparing the integers to rationals, there are obviously a wide
> variety of ways to qualitatively compare infinite sets of numbers, or
> other items.
Two ways come to mind: cardinality and Lebesgue measure.
> In terms of some eventual ultimate space of numbers, the
> big continuum, that's a universal number space, that's right number
> space, of sorts, and to be plain it is not addressed by standard
> theories.
But perhaps it could be addressed by some "ultimate universal number
space theory"? What kind of numbers would it cover?
> So, Dave, Dave, Dave, Steve, I was pointing out these things because
> Virgil was insulting you, and himself, and everybody else.
He's never insulted me.
David R Tribble
>> Half of the integers are even.
>
David R Tribble wrote:
>> True.
>
Jan Burse wrote:
> No.
> Here is my proof, 1/3 of the integers are even.
> Take the sequence:
> x=1 3 2 5 7 4 9 11 6 etc..
>
> Now cn = number of xi i<n and xi even.
> Lim n->oo cn/n = 1/3
>
> Proof:
> c1=0
> c2=0
> c3=0.33...
> c4=0.25
> c5=0.2
> c6=0.33...
>
> c3*k+1=k/(3*k+1)
> c3*k+2=k/(3*k+2)
> c3*k+3=(k+1)(3*k+3)
>
> Lim k->oo k/(3*k+1) = 1/3
> Lim k->oo k/(3*k+2) = 1/3
> Lim k->oo (k+1)(3*k+3) = 1/3
>
> So there is a single "häuffungspunkt" which is 1/3.
Perhaps we should consider the sequence:
x = 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, ...
This sequence can be used to show that 2/3 of the (positive) integers
are even.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/15/2006 12:54 AM, Virgil wrote:
> > In article <4559BAF3...@et.uni-magdeburg.de>,
> > Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> >> Following Galilei, I say:
> >> There are not more reals than rationals.
> >
> > When and where did Galileo Galilei lead that justifies anyone ever to
> > say anything like that?
>
> Galileo Galilei was arrested. Certainly you know why. So he had time to
> thoroghly think.
GG was not arrested for saying "There are not more reals than rationals",
and I find no evidence that he ever said anything like that.
Virgil
I have taken Ross to task for making claims he could not substantiate,
so that Ross may feel insulted.
David Marcus
I seem to recall that Galileo discovered that line segments of different
lengths could have their points put in one-to-one correspondence. Since
he thought it obvious that a longer segment must have more points, he
decided that it didn't make sense to talk about how many points there
were.
Of course, this doesn't really explain why EB thinks Galileo is
relevant.
--
David Marcus
Virgil
David Marcus <David...@alumdotmit.edu> wrote:
All the points of any segment must be equally real (or equally rational)
according to Galileo's discovery, so why EB thinks that discovery should
differentiate between reals and rationals will remain an unsolvable
mystery of the ages.
Ross A. Finlayson
Oh, I do, but that's not why. Pointing out some truth I would not find
an insult, calling everybody stupid, is insulting. Your manner changes
so much from person to person, but varies so little, constant yet
inconstant you are like a paradox, but just an annoyance.
Basically Hancher said that anybody who'd never heard of transfinite
cardinals is stupid. I find that insulting and wrong. As well, he
said there were no other ways to compare the "size" of two infinite
sets than by their cardinal and had only pejoratives for any who
thought otherwise. Obviously, Hancher has a pejorative mindset.
In terms of "meaningful differences" in some set's sizes, relative
differences, half the integers are even. What is the relative size of
those sets, in any superset?
Say all the choice functions over the integers can be uniformly
randomly sampled as a real number between negative one and one. What's
the probability that the first item is even? It's one half. Say
there's a random integer or arbitrarily large bound, essentially
unbounded, that's basically random, from the integers. What odds would
a reasonable wagerer accept that it was an even integer?
The density of the powers of four in the powers of two is one half, but
the density of the powers of two in the powers of one is zero, or
perhaps one if you take the zero'th power of two to be a power of one,
where the integer powers of one is a finite set. The density of powers
of coprimes, or multiples of one by yet another coprime, in each other
is zero, etcetera. It's easy to find reasonable asymptotic densities
like those. The probability of two integers being coprime is pi^2 / 6,
it is said.
As sets OF NUMBERS all the properties of being sets OF NUMBERS apply.
There is a framework for these numbers regardless of how they are
represented as sets. Half the integers are even, of all the numbers
that are integers, half of them are even.
In terms of measure, how can two intervals of measure one have measure
two? It's because they do not because the transfinite cardinals are
mute on the matter except surely saying nothing as they are continuous
line segments. It's consistent in ZF for the reals to have any of a
wide variety of cardinals. So, which one is it? There are
non-standard measure theories that quite happily go about analysis
without any transfinite cardinals.
A proper subset, of a regular set, is smaller than the set, in a sense
of "size." If you disagree you should write MIT and tell them they
awarded a wrong Ph.D.
(I'm a poet, so they tell me.)
About N E N and otherwise consideration of infinite sets being
irregular, there are some good reasons why in the non-pathological
cases that the limit can be considered the sum, that there is the point
at infinity, infinity in the naturals, as has been explored and
recorded since Gauss and so forth, who suggested avoiding sums of
infinite series, as he went about finding them. (To Galileo is
basically attributed that infinite sets are equivalent, where before
that there was instead consideration of the densities of the sets of
numbers, in number theory.) There is infinity in number theory.
Consider the sideways eight, lazy eight, the ouroboros, the lemniscate
as Wallis introduced it: it reads INFINITY.
Ross
David Marcus
> Virgil wrote:
> > In article <1163798916....@j44g2000cwa.googlegroups.com>,
> > "David R Tribble" <da...@tribble.com> wrote:
> > > > So, Dave, Dave, Dave, Steve, I was pointing out these things because
> > > > Virgil was insulting you, and himself, and everybody else.
> > >
> > > He's never insulted me.
> >
> > I have taken Ross to task for making claims he could not substantiate,
> > so that Ross may feel insulted.
>
> Oh, I do, but that's not why. Pointing out some truth I would not find
> an insult, calling everybody stupid, is insulting.
Has Virgil called anyone stupid who isn't stupid?
> Your manner changes
> so much from person to person, but varies so little, constant yet
> inconstant you are like a paradox, but just an annoyance.
--
David Marcus
Jan Burse
The oo is a good example for a sign that has
undergone a certain development over time, in
that it has been replaced by a set of more
differentiated signs.
Compare this process with your ability to name
colors. Maybe you will call something red simply
red, but compare this to somebody working for
example in textile manifacturing, he will be
able to enumerate million of names for red.
But I am not interested in the various signs
for cardinals and the like here.
The question is what meaning has remained
in oo and what meaning has been moved to other
symbols. As I see it the meaning that has
remained in oo is rather technical and is rather
a convenience of writing, than a deep insight.
The meaning that has remained in oo is the
handyness of having a uniform schema for
the denotation of intervals. And is comparable
to the ellipse, i.e. (...). Something like:
[2,3] = = { x | 2<=x & x<=3 }
[2,...] = [2,oo] = { x | 2<=x & x<=oo } = { x | 2<=x }
[...,2] = [-oo,2] = { x | -oo<=x & x<=2 } = { x | x<=2 }
This handyness comes with a big drawback. Namely
in an expression like x<=oo we switch domains.
We extend the relation <= originally defined on
reals, to a relation on the set of reals extended
by the two points oo and -oo.
This is in my view a certain uglyness, namely
because <= gets polymorphos, AND unnecessarely
a point is introduced in a set, that is neither
really used in the assumptions, the conclusions
or the proofs. It is also not used if you go
algebraic geometry.
If you look at the oo free definitions of intervals
you see that you can completely get away without
the symbol oo, i.e. you can reformulate it in terms
of the comparison operator that works on the reals
without the points oo and -oo.
So it is a fault of the lazyness of the math teachers,
and in consequence of the computer algebra system
developers, that the oo sign is still in use. Because
they think it is handy what they do. A deseases that
has even carried over into ordinals in set theory to
some degree. And computer science people fall also
into this trap. So one could see definitions of
open intervals like:
[2,...] = [2,Integer.MAX_VALUE]
Which works to a certain degree because in computers
the values are not unlimited. But if we for example
switch to big numbers, it becomes already a little ugly,
and maybe a pain for the memory of the computer, to
say something like:
[2,...] = [2,BigNumber.MAX_VALUE]
Will this bignumber eat up all memory of the computer,
and how can we continue to do something usefull after
all the memory has been eaten up?
So the hope is, that in the future the symbol oo will
slowly get away. That computer algebra introduce more
reflected denotations than the oo sign. And that they
will be able to work better with logic, for example
in that the fusion between computer algebra and constraint
solving will gain more ground, also in schools.
For a constraint solving system it is no problem to
introduce a constraint 2<=x, which perfectly reflects
the circumstances of x in [2,..].
Bye
David Marcus
> The meaning that has remained in oo is the
> handyness of having a uniform schema for
> the denotation of intervals. And is comparable
> to the ellipse, i.e. (...). Something like:
>
> [2,3] = = { x | 2<=x & x<=3 }
> [2,...] = [2,oo] = { x | 2<=x & x<=oo } = { x | 2<=x }
> [...,2] = [-oo,2] = { x | -oo<=x & x<=2 } = { x | x<=2 }
>
> This handyness comes with a big drawback. Namely
> in an expression like x<=oo we switch domains.
> We extend the relation <= originally defined on
> reals, to a relation on the set of reals extended
> by the two points oo and -oo.
>
> This is in my view a certain uglyness, namely
> because <= gets polymorphos, AND unnecessarely
> a point is introduced in a set, that is neither
> really used in the assumptions, the conclusions
> or the proofs. It is also not used if you go
> algebraic geometry.
It is handy in real analysis to allow functions to take values in the
extended real numbers.
--
David Marcus
Jan Burse
David Marcus wrote:
> It is handy in real analysis to allow functions to take values in the
> extended real numbers.
And problematic, respectively you only
get half way. There are operations that
are undefined, like:
0 * oo
oo
--
oo
etc..
As in:
1 x
lim --*x = lim -- = 1/2
x->oo 2x x->oo 2x
You cannot evaluate them on the
extended real line. As:
lim f(x) = f(oo) only if f(oo) defined
x->oo
You can only make them defined,
when you go more algebraic.
Bye
See also: http://en.wikipedia.org/wiki/Infinity#Undefined_operations
Ross A. Finlayson
> Ross A. Finlayson wrote:
> > Virgil wrote:
> > > In article <1163798916....@j44g2000cwa.googlegroups.com>,
> > > "David R Tribble" <da...@tribble.com> wrote:
> > > > Ross A. Finlayson wrote:
>
> > > > > So, Dave, Dave, Dave, Steve, I was pointing out these things because
> > > > > Virgil was insulting you, and himself, and everybody else.
> > > >
> > > > He's never insulted me.
> > >
> > > I have taken Ross to task for making claims he could not substantiate,
> > > so that Ross may feel insulted.
> >
> > Oh, I do, but that's not why. Pointing out some truth I would not find
> > an insult, calling everybody stupid, is insulting.
>
> Has Virgil called anyone stupid who isn't stupid?
>
> > Your manner changes
> > so much from person to person, but varies so little, constant yet
> > inconstant you are like a paradox, but just an annoyance.
>
> --
> David Marcus
Apparently so, yeah, Virgil regularly insults people in his ad hominem
rantings.
He called EVERYBODY stupid. So, is anybody not? "On the idiot scale,
even Feldmann's an idiot."
Hauling out standardized test scores doesn't help me. (It doesn't hurt
me.)
I find of interest the N^N, that is NxNxNx..., the product of
infinitely many copies of N, interesting here in that it's consistently
equivalent to the reals, say. This is where it is said that there is a
uniform probability distribution over, say, the unit interval of reals
and thus correspondingly these N-tuples, infinite sequences of elements
of N. Then, I get to wondering about the subset of those sequences
that have only one of each element, ie, the choice functions of N.
Then it seems it would take R^N many samples to get one of those
sequences, and, that is more than there are. That is where almost none
of the elements of N^N, N-tuples, have no duplicated elements, where
each initial segment has no duplicates.
That's digression, it seems the choice sequences would be unformly
distributed within the N-tuples, for no element of an initial segment
is repeated, and that there would be many samples of the N-tuples to
get an N-choice sequence. When you did get an N-choice sequence, from
the uniformly random real, the first item would thus be a uniformly
random natural integer, thus, where the above is so, there is a uniform
probability distribution over the naturals, or its existence where
those other things exist.
Back to the beginning of this thread, one of the supertasks as
described on the Wiki page is mentioned, the Zeno relay race of sorts,
phrased in the context of total impart of momentum in eventually
densely spaced bearings of the executive desk toy, for no last element
of the natural integers. There are infinitely many runners spaced at
0, 1/2, 3/4, 7/8, etcetera on the track in the race between zero and
one, runner one carries a baton and passes it in his constant velocity
progress when he reaches runner two, who proceeds apace of a pace
runner who simply moves at the same constant velocity between zero and
one. The pace runner moves at constant velocity, the relay runners are
never ahead nor behind. Does Zeno ever get anywhere? Then the sum is
the limit is the sum, in the infinite, in that case.
Ross
David Marcus
> David Marcus wrote:
> > It is handy in real analysis to allow functions to take values in the
> > extended real numbers.
>
> And problematic, respectively you only
> get half way.
But, why is that a problem? And, halfway to what?
> There are operations that
> are undefined, like:
>
> 0 * oo
In measure theory, the convention is to define 0 * oo = 0 (see, e.g.,
"Real Analysis" by H.L. Royden or your wikipedia reference). Again, this
is handy.
> oo
> --
> oo
> etc..
>
> As in:
> 1 x
> lim --*x = lim -- = 1/2
> x->oo 2x x->oo 2x
>
> You cannot evaluate them on the
> extended real line. As:
>
> lim f(x) = f(oo) only if f(oo) defined
> x->oo
>
> You can only make them defined,
> when you go more algebraic.
>
> Bye
>
> See also: http://en.wikipedia.org/wiki/Infinity#Undefined_operations
--
David Marcus
David Marcus
> David Marcus wrote:
> > Ross A. Finlayson wrote:
> > > Virgil wrote:
> > > > In article <1163798916....@j44g2000cwa.googlegroups.com>,
> > > > "David R Tribble" <da...@tribble.com> wrote:
> > > > > Ross A. Finlayson wrote:
> >
> > > > > > So, Dave, Dave, Dave, Steve, I was pointing out these things because
> > > > > > Virgil was insulting you, and himself, and everybody else.
> > > > >
> > > > > He's never insulted me.
> > > >
> > > > I have taken Ross to task for making claims he could not substantiate,
> > > > so that Ross may feel insulted.
> > >
> > > Oh, I do, but that's not why. Pointing out some truth I would not find
> > > an insult, calling everybody stupid, is insulting.
> >
> > Has Virgil called anyone stupid who isn't stupid?
> >
> > > Your manner changes
> > > so much from person to person, but varies so little, constant yet
> > > inconstant you are like a paradox, but just an annoyance.
>
> rantings.
Are you sure he isn't saying that what the person writes is stupid?
> He called EVERYBODY stupid.
David Tribble says Vigil has never insulted him, so I assume that means
he's never called him stupid. Seems like you are the one insulting
Virgil.
> So, is anybody not? "On the idiot scale, even Feldmann's an idiot."
--
David Marcus
Virgil
David Marcus <David...@alumdotmit.edu> wrote:
David Marcus, David C. Ulrich, Arturo Magidin, Christopher Heckman, as
well as David Tribble, are among the many others whom I have not called
stupid.
It is an honor only won by those exhibiting considerable perseverence in
earning it, so I do not award it often.
IIRC, the first honoree was JSH, whom everyone will acknowledge has
perseverence enough.
Ross A. Finlayson
The point is that Feldmann's not an idiot.
No, it just means he chooses to ignore it, Virgil's regular ad hominem,
against the man, gaffes. There is much to see in a man in how he
writes.
As you can see, what here could have been interesting discussion about
the infinite and foundations of mathematics has descended into who
insulted who when. Why?
So, do you think that's not an existence proof of a uniform probability
distribution over the natural integers?
If a proper subset has via a defined size relation a smaller size, is
that not so?
How about the universe? Where functions between objects are objects
and the set of functions between copies of a set of objects bijects to
the powerset of the set of objects, is not the physical universe an
example that infinite set and powerset are equivalent, via identity?
How about Zeno's supertask? Does the racer match the relay, and vice
versa, or not?
Please specifically and unambiguously answer each question.
Ross
David Marcus
>
> No, it just means he chooses to ignore it, Virgil's regular ad hominem,
> against the man, gaffes. There is much to see in a man in how he
> writes.
>
> As you can see, what here could have been interesting discussion about
> the infinite and foundations of mathematics has descended into who
> insulted who when. Why?
Because you said Virgil was insulting people?
> So, do you think that's not an existence proof of a uniform probability
> distribution over the natural integers?
>
> If a proper subset has via a defined size relation a smaller size, is
> that not so?
>
> How about the universe? Where functions between objects are objects
> and the set of functions between copies of a set of objects bijects to
> the powerset of the set of objects, is not the physical universe an
> example that infinite set and powerset are equivalent, via identity?
>
> How about Zeno's supertask? Does the racer match the relay, and vice
> versa, or not?
>
> Please specifically and unambiguously answer each question.
Please ask a specific and unambiguous question.
--
David Marcus
Ross A. Finlayson
Yeah, that's why.
If you can select at uniform random from the choice-sequences of N, is
not the first element of a sample an element at uniform random from N?
If a proper subset is smaller is it not, in the very same sense,
smaller?
Is there a universe? Is there not?
Is there a point at infinity? Does anything ever happen?
These are the same questions.
Ross
David Marcus
> David Marcus wrote:
> > Ross A. Finlayson wrote:
> > > The point is that Feldmann's not an idiot.
> > >
> > > No, it just means he chooses to ignore it, Virgil's regular ad hominem,
> > > against the man, gaffes. There is much to see in a man in how he
> > > writes.
> > >
> > > As you can see, what here could have been interesting discussion about
> > > the infinite and foundations of mathematics has descended into who
> > > insulted who when. Why?
> >
> > Because you said Virgil was insulting people?
> >
> > > So, do you think that's not an existence proof of a uniform probability
> > > distribution over the natural integers?
> > >
> > > If a proper subset has via a defined size relation a smaller size, is
> > > that not so?
> > >
> > > How about the universe? Where functions between objects are objects
> > > and the set of functions between copies of a set of objects bijects to
> > > the powerset of the set of objects, is not the physical universe an
> > > example that infinite set and powerset are equivalent, via identity?
> > >
> > > How about Zeno's supertask? Does the racer match the relay, and vice
> > > versa, or not?
> > >
> > > Please specifically and unambiguously answer each question.
> >
> > Please ask a specific and unambiguous question.
>
>
> If you can select at uniform random from the choice-sequences of N, is
> not the first element of a sample an element at uniform random from N?
>
> If a proper subset is smaller is it not, in the very same sense,
> smaller?
>
> Is there a universe? Is there not?
>
> Is there a point at infinity? Does anything ever happen?
>
> These are the same questions.
I meant: Please ask a specific, unambiguous, and mathematically sensible
question.
--
David Marcus
Ross A. Finlayson
Well, here are some answers.
If you can select at uniform random from the choice sequences of N, via
from the sequences of elements of N, as can be done from the (finite
interval of) reals, then, you can from N.
The proper subset is smaller, in size, than the set, where that here on
sci.math is attributed.
There is a universe, it's infinite, and an infinite set and powerset
are thusly shown equivalent, the universe.
Sum the differentials between zero and one.
Yeah, that's why.
Ross
Eckard Blumschein
> you believe you have some reasonable argument
> against Set Theory as it is now.
Arguments rather than a single argument. All of them seem to be serious.
On the other hand, the swarm of counterarguments seem to not contain any
really serious scientific one, just 1.000.000 Mark, some rather populist
suggestions, untennable statements, arbitrarily chosen axioms, putative
wisdom of the majority, etc.
>> Well, I am proud of my teacher of mathematics Prof. Nikolaus Joachim
>> Lehmann who was simultaneously a German pioneer of "small" computers
>> when I studied Electrical Engineering at University of Dresden from 1960
>> to 1966. I will never forget him climbing from his blackboard beneath of
>> the auditory, almost jumping from desk to desk until he reached the
>> entrance on top. The auditory was overcrowded because Nabla J., as we
>> called him, had overrun his time limit. We were still sitting behind our
>> desks while the next students pushed into the auditory from above. Two
>> of them insulted him. Imagine, he stood beneath at the blackboard. They
>> felt safe at the pretty distant entrance above. No matter for a
>> personality like Nabla J, no matter that he was not slim. The two
>> students could not escape, since there was a bulk of pushing students
>> behind them. Nabla J. captured them, climbing over the desk we were
>> sitting behind.
> *****************************************************************************
> Great athlete, apparently, that J. Lehmann professor. As mathematician
> I can't say anything: I just don't know him
In order to look into his face you may google for "Nikolaus Joachim
Lehmenn".
His research in what we would call PCs was stopped by narrow-minded
experts of party and government of GDR in the early sixtees.
Eckard Blumschein
>> GG was not arrested for saying "There are not more reals than rationals",
Everybody knows the famous sentence related to the rotation of earth:
(In German: Und sie dreht sich _doch_!)
I do not know an appropriate translation, you will understand:
It indeed rotates!
>> and I find no evidence that he ever said anything like that.
>
> I seem to recall that Galileo discovered that line segments of different
> lengths could have their points put in one-to-one correspondence.
Yes. Even Cantor took advantage of this method.
> Since
> he thought it obvious that a longer segment must have more points, he
> decided that it didn't make sense to talk about how many points there
> were.
He understood that there is a logical contradiction which can only be
resolved if one accepts that the relations smaller than, equally many
and more than are invalid if one tries to compare with each other
infinite quantities.
> Of course, this doesn't really explain why EB thinks Galileo is
> relevant.
Galilei was perhaps the first one but definitely not the only one who
understood this 4th possibility: incomparability.
Eckard Blumschein
On 11/18/2006 11:51 PM, Virgil wrote:
> All the points of any segment must be equally real (or equally rational)
> according to Galileo's discovery, so why EB thinks that discovery should
> differentiate between reals and rationals will remain an unsolvable
> mystery of the ages.
I understand your lack of understanding in this case, too: Someone who
did not yet grasp that the rationals are genuine numbers but the reals
are fictions ones cannot comprehend that both reals and rationals are
infinite and nonetheless quite different from each other.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/17/2006 11:07 AM, Tonico wrote:
>
> > you believe you have some reasonable argument
> > against Set Theory as it is now.
>
> Arguments rather than a single argument. All of them seem to be serious.
To their author(s), perhaps, but none show any internal inconsistency in
ZF or ZFC or NBG or NF, as all of them make other contrary assumptions
for which there is no valid basis.
The only valid opposition to any of these set theories is that you just
do not choose to accept them.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/18/2006 11:05 PM, David Marcus wrote:
>
> >> GG was not arrested for saying "There are not more reals than rationals",
>
> Everybody knows the famous sentence related to the rotation of earth:
> (In German: Und sie dreht sich _doch_!)
> I do not know an appropriate translation, you will understand:
> It indeed rotates!
>
>
> >> and I find no evidence that he ever said anything like that.
> >
> > I seem to recall that Galileo discovered that line segments of different
> > lengths could have their points put in one-to-one correspondence.
>
> Yes. Even Cantor took advantage of this method.
It has been used by just about every geometer since Euclid.
It shows, among other things, there are bijections between the set of
points in any line segment and the set of points between its midpoint
and one end.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
Every real and every rational is finite and they are equally fictitious
or equally actual.
It is only sets of them that can be otherwise than finite.
EB's sloppiness with mathematical terminology makes it impossible to
consider his claims seriously.
Tonico
Eckard Blumschein wrote:
> On 11/18/2006 11:05 PM, David Marcus wrote:
>
> >> GG was not arrested for saying "There are not more reals than rationals",
>
> Everybody knows the famous sentence related to the rotation of earth:
> (In German: Und sie dreht sich _doch_!)
> I do not know an appropriate translation, you will understand:
> It indeed rotates!
>
>
> >> and I find no evidence that he ever said anything like that.
> >
> > I seem to recall that Galileo discovered that line segments of different
> > lengths could have their points put in one-to-one correspondence.
>
> Yes. Even Cantor took advantage of this method.
>
>
> > Since
> > he thought it obvious that a longer segment must have more points, he
> > decided that it didn't make sense to talk about how many points there
> > were.
>
> He understood that there is a logical contradiction which can only be
> resolved if one accepts that the relations smaller than, equally many
> and more than are invalid if one tries to compare with each other
> infinite quantities.
Nonsense. GG was a man of his time (who isn't?) and hadn't the powerful
resources Cantor, Dedekind, Hilbert, etc. had to their dispossal. The
"logical contradiction" that you say GG understood (how do you know
what he understood? Are you into ouija and stuff like that?) is not
more "logical" than the ""logical deduction"" by Aristotle that heavier
bodies must fall faster to the earth when dropped...
*****************************************************
> > Of course, this doesn't really explain why EB thinks Galileo is
> > relevant.
>
> Galilei was perhaps the first one but definitely not the only one who
> understood this 4th possibility: incomparability.
???
Regards
Tonio
Tonico
Eckard Blumschein wrote:
> On 11/17/2006 11:07 AM, Tonico wrote:
>
> > you believe you have some reasonable argument
> > against Set Theory as it is now.
>
> Arguments rather than a single argument. All of them seem to be serious.
>
> On the other hand, the swarm of counterarguments seem to not contain any
> really serious scientific one, just 1.000.000 Mark, some rather populist
> suggestions, untennable statements, arbitrarily chosen axioms, putative
> wisdom of the majority, etc.
You talk of populism?!?! Hehe...oooo-kaaaaay. And, of course, you get
to decide what's "scientific" and what isn't.....aha. "Putative wisdom
of the majority"...I bet you meant to say something with this. Ah, and
axioms, my boy, are MANY times arbitrary: what's wrong with this, in
your opinion? Don't like some set of axioms? Fine, propose yours....and
let's give it some time to check whether they're consistent, just as ZF
has proved to be so far.
I'm guessing that since you are, apparently, an ex-DDR (democratic
republic of Germany)'s citizen, you think it is very cool to go against
"majority", without checking too deep whether that "majority" is
REALLY right or not in some respect. I can understand that feeling in
people from former dictatorial countries, but hey: that doesn't make
your arguments any better than others'. Believe me, it REALLY does not.
Tonio
***********************************************************
Bob Kolker
> On 11/18/2006 11:05 PM, David Marcus wrote:
>
> Everybody knows the famous sentence related to the rotation of earth:
> (In German: Und sie dreht sich _doch_!)
> I do not know an appropriate translation, you will understand:
> It indeed rotates!
Eppur si mouve! What Galileo was supposed to have said under his breath
after he abjured.
And it -does- turn (or move).
Bob Kolker
Bob Kolker
>
> I understand your lack of understanding in this case, too: Someone who
> did not yet grasp that the rationals are genuine numbers but the reals
> are fictions ones cannot comprehend that both reals and rationals are
> infinite and nonetheless quite different from each other.
All numbers are fictitious. They exist only in our thoughts along with
dragons and unicorns.
Bob Kolker
>
Ross A. Finlayson
Is the universe real, then, Bob?
Here, Bob is quite the opinionist, and here's his stated opinion about
the universe being an example that infinite sets are equivalent: no
opinion.
So, is there is a universe?
Thanks,
Ross
David Marcus
> On 11/18/2006 11:05 PM, David Marcus wrote:
> > I seem to recall that Galileo discovered that line segments of different
> > lengths could have their points put in one-to-one correspondence.
>
> Yes. Even Cantor took advantage of this method.
>
> > Since
> > he thought it obvious that a longer segment must have more points, he
> > decided that it didn't make sense to talk about how many points there
> > were.
>
> He understood that there is a logical contradiction which can only be
> resolved if one accepts that the relations smaller than, equally many
> and more than are invalid if one tries to compare with each other
> infinite quantities.
Please tell us what this logical contradiction is.
--
David Marcus
Jan Burse
David Marcus wrote:
> Jan Burse wrote:
> In measure theory, the convention is to define 0 * oo = 0 (see, e.g.,
> "Real Analysis" by H.L. Royden or your wikipedia reference). Again, this
> is handy.
Did you see 0 * oo = 0 in my example? Here
is again my example:
1
lim --*x = 0 * oo = 1/2
x->oo 2x
Stop reading wikipedia, start thinking by yourself.
David Marcus
> David Marcus wrote:
What's your point? The limit as x -> oo isn't calculated by substituting
oo for x.
--
David Marcus
Ross A. Finlayson
So, how are you intelligent people with a sophisticated background in
mathematics doing with answers to those questions?
Don't you think the trolls here would jump at the chance to answer
these questions if it didn't expose flaws in their stated opinions?
You bet they would. They'd be happy to do so.
So, do the answers to these questions expose flaws in their stated
opinions?
Ross
David Marcus
> Ross A. Finlayson wrote:
> > David Marcus wrote:
> > > Ross A. Finlayson wrote:
> > > >
> > > > If you can select at uniform random from the choice-sequences of N, is
> > > > not the first element of a sample an element at uniform random from N?
> > > >
> > > > If a proper subset is smaller is it not, in the very same sense,
> > > > smaller?
> > > >
> > > > Is there a universe? Is there not?
> > > >
> > > > Is there a point at infinity? Does anything ever happen?
> > > >
> > > > These are the same questions.
> > >
> > > I meant: Please ask a specific, unambiguous, and mathematically sensible
> > > question.
> >
> >
> > If you can select at uniform random from the choice sequences of N, via
> > from the sequences of elements of N, as can be done from the (finite
> > interval of) reals, then, you can from N.
> >
> > The proper subset is smaller, in size, than the set, where that here on
> > sci.math is attributed.
> >
> > There is a universe, it's infinite, and an infinite set and powerset
> > are thusly shown equivalent, the universe.
> >
> > Sum the differentials between zero and one.
> >
> > Yeah, that's why.
>
> So, how are you intelligent people with a sophisticated background in
> mathematics doing with answers to those questions?
Still waiting for the specific, unambiguous, and mathematically sensible
questions I asked for.
> Don't you think the trolls here would jump at the chance to answer
> these questions if it didn't expose flaws in their stated opinions?
> You bet they would. They'd be happy to do so.
You jumped at the chance to answer the questions. Did you do that
because you are a troll?
> So, do the answers to these questions expose flaws in their stated
> opinions?
No. But the questions (and your answers) certainly expose flaws in the
questioner.
--
David Marcus
Ross A. Finlayson
Hi Dave,
No, apparently not, because they don't conflict with my stated
opinions.
Troll. That's not fair, you obviously have an interest in this kind of
thing even if you go about it in a troll-, troll-like? troll-ific,
troll-erian, troll-, in trollery.
There is no universe in ZF, and according to ZF nothing that could ever
be its universe, so it can't quantify yet it must so the soi-disant set
theory ZF contradicts itself, in ZF.
That at least has a more or less well-established response, that the
cumulative hierarchy exists and can model any system of interest, in
ZF. The problem with that is that the cumulative hierarchy isn't in
ZF.
About the relay and runner, and the runner is the relay, with nods to
the transfer principle, there is a resolution and it's basically an
alternative to the Russellian branch taken about a hundred years ago,
and myriad classical and modern treatments support it.
About the proper subset size relation on a set, I have removed some
material here, there's less.
Those are old hat, the question in which I have particular interest at
this time is if via a method to select at uniform random an element or
infinitely many elements of the real numbers whether via a pretty
simple construction that can be construed as a method to select a
natural integer at uniform random, as I have so described.
If you're not going to answer those questions I will. So, who's the
questioner?
Ross
Eckard Blumschein
Tonico
Do you really think (figure of speech) that any stupidity deserves an
answer, son? I bet you don't...right?
Good!
Tonio
Eckard Blumschein
Since there are many competing religions, maybe not a single one is the
true one.
Since there are many competing set theories, this fits to the result of
my look into pertaining literature: Neither of these "theories" has a
proven basis.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> I give in. Vergil is too stupid.
Certainly too stupid to be persuaded into believing falsehoods by a fool.
Tonico
Eckard Blumschein wrote:
> I give in. Vergil is too stupid.
Is that the reason you're giving up?!? Bah, and I thought you realized
FINALLY that YOUR line of argumentation was stupid....oh, well: as long
as you REALLY gave up.
Tonio
Eckard Blumschein
> Don't like some set of axioms? Fine, propose yours....and
> let's give it some time to check whether they're consistent,
Archimede on the notion number
Euclid on the notion point
Peirce on the notion continuum
Spinoza on the notion infinite
Check them please.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
Axiom systems do not have "proven" bases because we have no access to
any "absolute" truths that are free of any assumptions whatsoever.
To do anything we must make assumptions. it behooves us to make them
wisely.
I find EB's unproven assumptions frivolous, deleterious and occasionally
mutually contradictory.
Virgil
Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
> On 11/20/2006 10:28 AM, Tonico wrote:
> > Don't like some set of axioms? Fine, propose yours....and
> > let's give it some time to check whether they're consistent,
>
> Archimede on the notion number
> Euclid on the notion point
> Peirce on the notion continuum
> Spinoza on the notion infinite
>
> Check them please.
>
Can EB assure us that each of these persons has one and only one
statement on the relevant issue?
Can EB assure us that the statements form these 4 people on these 4
items form a complete set of axioms for EB's version of mathematics,
other than those of standard formal logic, and may we presume that EB
accepts standard formal logic?
Tonico
Eckard Blumschein wrote:
> On 11/20/2006 10:28 AM, Tonico wrote:
> > Don't like some set of axioms? Fine, propose yours....and
> > let's give it some time to check whether they're consistent,
>
> Archimede on the notion number
> Euclid on the notion point
> Peirce on the notion continuum
> Spinoza on the notion infinite
>
> Check them please.
Pssst, I don't know....since you're going to be REALLY serious and
propose to the mathematical community some new axioms in order to...to
whatever, I think it'd be just as serious to require from you to
EXPLICITY write down those axioms. What is "Archimede notion of
number"?? I don't have the faintest idea, and I think I know something
about Archimede and the notion of number: what do you mean EXACTLY? Do
the same with Euclide and all the other OLD, OLD great men mentioned by
you.
After you do the above, it'd be nice if you could point out how those
new axioms of yours should complement, or change, some actual axioms in
maths: for example, what will Euclide's notion of point, whatever that
is (and I supose it means the beginning of the first book of the
Elements, where a point is assumed to be a rather ethereal entity, some
kind of "primitive notion in every one of us", without volume, mass,
length or any other physical measures), and how this could POSSIBLY
affect modern maths. I can't see, for example, that Euclide could
affect Dedekind-Cantor's set theory in any way....but who knows?
Perhaps you've some insights that we haven't and you can FORMALIZE
those insights.
Tonio
Heinz Mau
> Archimede on the notion number
> Euclid on the notion point
> Peirce on the notion continuum
> Spinoza on the notion infinite
You are as thick as two short planks.
David Marcus
> I give in. Vergil is too stupid.
Or, not stupid enough.
--
David Marcus
Ross A. Finlayson
> Those are old hat, the question in which I have particular interest at
> this time is if via a method to select at uniform random an element or
> infinitely many elements of the real numbers whether via a pretty
> simple construction that can be construed as a method to select a
> natural integer at uniform random, as I have so described.
>
About this notion of a bijection between the unit interval of reals and
N^N, in standard ways, and about some informal consideration of the
N-choice sequences or in shorthand N-sequences being uniformly dense
within the elements of N^N, I should enumerate an obvious progression
of reasoning to that effect.
Basically the sketch is this: the reals in the unit interval biject to
N^N, the cartesian product of infinitely many copies of the natural
integers. There exists a uniform probability distribution over the
unit interval of reals, and thus the elements of N^N, in that each
element of N^N so correlated, informally, has an equal probability of
being selected as a sample as does any element in R[0,1].
Now, among all those elements of N^N, very very few of them are
N-sequences, sequences with each element of the natural integers being
an element of the sequence exactly once, where those are ranges of
choice functions, well-orderings of the natural integers and have a
least, in this case first, element of the sequence. None of those
sequences are identical. So, sample from N^N at uniform random and
discard the sample if it is not an N-sequence, and sample until there
is an N-sequence, selected at uniform random from among the
N-sequences.
Then, each N-sequence in N^N has the same probability of being
selected. As samples of the natural integers, let the N-sequences that
begin with n be N(n)-sequences, N sequences that begin with x, y, z
being N(x, y, z)-sequences, for and etcetera, for a finite number of
variables. There is an N(n)-sequence for each n, finite natural
integer in N.
The N(n)-sequences can be considered identical, the elements of the
sequence after the first are inconsequential.
Then that seems to beg the question of there being a natural uniform
probability distribution over the natural integers for there to be one.
While that may be so, each of the N(n)-sequences has the same
probability of being selected as any other, as each is an element of
N^N. So, it may be disregarded which one it is.
I'm not sure about bijecting R[0,1] to N^N, but it seems that if it
took R^N to biject to N^N then R would biject to N. Otherwise there
would be cardinals between those of N and P(N).
Taking infinitely many samples of R to sample N^N is the same as
sampling infinitely many samples of {0,1} to sample R, and that returns
infinitely many samples. That process returning a rational would
return infinitely many copies of a variety of rationals, and returning
an algebraic irrational would seem to return only algebraics.
Returning zero means all zeros.
So, yeah, that's why.
Ross
Eckard Blumschein
Did you read the pamphlet by Christian Betsch? He was paid 1,000,000.00
Mark which was a huge amount of money in 1926 after inflation.
Why? He nurtured just this demagogic line of argumentation. The aim was
quite clear: To protect the apriorism by Dedekind and Cantor from any
attack based on unbiased reasoning.
Eckard Blumschein
Eckard Blumschein
> GG was a man of his time (who isn't?) and hadn't the powerful
> resources Cantor, Dedekind, Hilbert, etc. had to their dispossal.
What about Cantor, he read only a few important papers and these for the
first and perhaps only time each time he was in a mad house where he did
not have obligations like a teacher.
Even Hilbert did just have libraries to his disposal.
> The
> "logical contradiction" that you say GG understood (how do you know
> what he understood?
Galilei wrote a fictitious discurse between three people: thesis,
antithesis, synthesis.
Heinz Mau
> Did you read the pamphlet
Please keep moving. Thats only our little stupid girl from germany.
Sorry that we let him out.
Eckard Blumschein
Thank you.
Eckard Blumschein
> Eckard Blumschein wrote:
>> Since there are many competing set theories, this fits to the result of
>> my look into pertaining literature: Neither of these "theories" has a
>> proven basis.
>
> Axiom systems do not have "proven" bases because we have no access to
> any "absolute" truths that are free of any assumptions whatsoever.
>
> To do anything we must make assumptions. it behooves us to make them
> wisely.
Imagine someone who has made progress step by step over long time
telling you he suddenly managed to reach within a comparatively tiny
timespan much more than in all time before, would you believe him?
> I find EB's unproven assumptions frivolous, deleterious
Deleterious to illusions, yes.
> and occasionally mutually contradictory.
Evidence or silence please.
Bob Kolker
>
> Why? He nurtured just this demagogic line of argumentation. The aim was
> quite clear: To protect the apriorism by Dedekind and Cantor from any
> attack based on unbiased reasoning.
He also was factually correct.
If there were no sentients in the cosmos there would be no numbers.
Bob Kolker
Bob Kolker
>
> Galilei wrote a fictitious discurse between three people: thesis,
> antithesis, synthesis.
And I thought it was Salviati, Simplicio and Sagredo.
Live and learn.
Bob Kolker
Eckard Blumschein
> In article <4562C74C...@et.uni-magdeburg.de>,
> Eckard Blumschein <blums...@et.uni-magdeburg.de> wrote:
>
>> On 11/20/2006 10:28 AM, Tonico wrote:
>> > Don't like some set of axioms? Fine, propose yours....and
>> > let's give it some time to check whether they're consistent,
>>
>> Archimede on the notion number
>> Euclid on the notion point
>> Peirce on the notion continuum
>> Spinoza on the notion infinite
>>
>> Check them please.
>>
> Can EB assure us that each of these persons has one and only one
> statement on the relevant issue?
I repetitiously wrote what I am referring to.
> Can EB assure us that the statements form these 4 people on these 4
> items form a complete set of axioms for EB's version of mathematics,
> other than those of standard formal logic, and may we presume that EB
> accepts standard formal logic?
I am not sure if mathematics as a whole really needs sets of axioms.
The devil resides in the details, in particular in the used termiology.
Formal logic, as excellent as it is, can easily be misused.
Verbal description is more concise:
Any number is basic to a subsequent one.
A point is what does not have parts.
A continuum is something every part of which has parts.
Infinity is the quality of being neither enlargeable nor exhaustable.
Check these definitions please.
While set theory fails because it uses the undefined term set, the four
definitions do not rely on questionable terminology.
Eckard Blumschein
On 11/21/2006 11:16 AM, Tonico wrote:
> Eckard Blumschein wrote:
>> On 11/20/2006 10:28 AM, Tonico wrote:
>> > Don't like some set of axioms? Fine, propose yours....and
>> > let's give it some time to check whether they're consistent,
>>
>> Archimede on the notion number
>> Euclid on the notion point
>> Peirce on the notion continuum
>> Spinoza on the notion infinite
>>
>> Check them please.
> *****************************************************
> What is "Archimede notion of
> number"??
The axiom of infinity is just a mutilated plagiat. Take it and remove
the added toxic cover, i.e. the term "set".
> .... what will Euclide's notion of point, ... affect modern maths.
The notions of point and number are basic to Cantor's "set of points" as
an aprioric surrogate of numbers.
> I can't see, for example, that Euclide could
> affect Dedekind-Cantor's set theory in any way....but who knows?
> Perhaps you've some insights that we haven't and you can FORMALIZE
> those insights.
Yesterday in the evening I heared for the first time in Magdeburg
someone on the street loudly telling the story of God to everybody.
Can you imagine anybody convincing him by means of formalized sentences
from the bible that other religions if any also might be correct?
I consider this believer at least as biased as you, Virgil and many many
others together.