Shakespeare and monkeys
John Schmitt
the monkeys-typewriters-Shakespeare theory by experiment. If you
go to their homepage and search using the obvious terms in the
near future, you can read an article which will, at the very
least, make the corners of you mouth turn upwards. It is also at
the BBC site:
http://news.bbc.co.uk/1/hi/england/devon/3013959.stm
Admittedly it is a very small sample, with hardly enough monkeys,
typewriters or time, but it is heartening to hear that bashing a
horse until only a faint stain remains is still a sport academia
supports.
John "a monkey barred" Schmitt
--
If you have nothing to say, or rather, something extremely stupid
and obvious, say it, but in a 'plonking' tone of voice - i.e.
roundly, but hollowly and dogmatically. - Stephen Potter
Jeff Lanam
joh...@alpha1.mdx.ac.uk (John Schmitt) emitted:
>Today's Times contains a report about an attempt to demonstrate
>the monkeys-typewriters-Shakespeare theory by experiment. If you
>go to their homepage and search using the obvious terms in the
>near future, you can read an article which will, at the very
>least, make the corners of you mouth turn upwards. It is also at
>the BBC site:
I heard a report this morning about this on CBS radio. They
referred to "scientists at the University of Plymouth," implying
that people with real academic credentials did it. The BBC
credits "lecturers and students", and quotes the Director of
the Institute for Digital Arts and Technology. The project
got 2000 pounds from the Arts Council. Sounds more like a
performance art piece than a scientific experiment.
ObUL: You know it's going to be cited by people as an example
of those crazy scientists wasting public money.
John Francis
Jeff Lanam <jeff.lana...@compaq-dot-com.not> wrote:
>On Fri, 9 May 2003 13:08:15 +0000 (UTC), the evil clone of
>joh...@alpha1.mdx.ac.uk (John Schmitt) emitted:
>
>>Today's Times contains a report about an attempt to demonstrate
>I heard a report this morning about this on CBS radio. They
>referred to "scientists at the University of Plymouth," implying
>that people with real academic credentials did it. The BBC
>credits "lecturers and students", and quotes the Director of
>the Institute for Digital Arts and Technology. The project
>got 2000 pounds from the Arts Council. Sounds more like a
>performance art piece than a scientific experiment.
According to the Guardian article that same day (Friday) the test
was designed by Geoff Cox of Plymouth University's MediaLab, paid
for by a 2,000 UKP grant from the Arts Council, and carried out
at Paignton Zoo in Devon.
A quote from Mr. Cox:
"It wasn't actually an experiment as such, it was more like a
little performance"
One I like even better, altough perhaps of less overall relevance:
"[Monkeys] get bored and they shit on the keyboard rather than type"
I feel we can all learn something from this.
--
As evil plans go, it doesn't suck -- Wesley offers a critique on "Angel"
R H Draney
>
>One I like even better, altough perhaps of less overall relevance:
>
>"[Monkeys] get bored and they shit on the keyboard rather than type"
>
>I feel we can all learn something from this.
Well, I'll be...they really *are* just like human authors....r
Hatunen
<dado...@earthlink.net> wrote:
Should David Ives' play, "All in the Timing" be performed
anywhere in the vicinity I recommend everyone to see it. One of
the scenes involves severaal monkeys which have been put to the
task of writing Shakespeare. Another scene involves a number of
variations on how Leon Trotsky came to have a pick stuck in his
head.
************* DAVE HATUNEN (hat...@cox.net) *************
* Tucson Arizona, out where the cacti grow *
* My typos & mispellings are intentional copyright traps *
Lee
>ObUL: You know it's going to be cited by people as an example
>of those crazy scientists wasting public money.
Since they're selling copies of their report on the Internet,
this may be an example of crazy scientists making a profit.
Simon Slavin
jo...@panix.com (John Francis) wrote:
>A quote from Mr. Cox:
>
>"It wasn't actually an experiment as such, it was more like a
>little performance"
>
>One I like even better, altough perhaps of less overall relevance:
>
>"[Monkeys] get bored and they shit on the keyboard rather than type"
>
>I feel we can all learn something from this.
How ? We're already /on/ usenet.
Phat Ratty Ratt
>Today's Times contains a report about an attempt to demonstrate
>the monkeys-typewriters-Shakespeare theory by experiment. If you
>go to their homepage and search using the obvious terms in the
>near future, you can read an article which will, at the very
>least, make the corners of you mouth turn upwards. It is also at
>the BBC site:
>
>http://news.bbc.co.uk/1/hi/england/devon/3013959.stm
>
>Admittedly it is a very small sample, with hardly enough monkeys,
>typewriters or time, but it is heartening to hear that bashing a
>horse until only a faint stain remains is still a sport academia
>supports.
True, they didn't write Shakespeare (presumably for lack of monkeys,
typewriters, and time), but they did produce the scripts for several new Fox
Reality TV shows and a couple of speeches for (insert politician's name here).
Ratt Boy, who has it on good authority that Ruppert Murdoch descended from an
ape.
Charles A Lieberman
jeff.lana...@compaq-dot-com.not (Jeff Lanam) wrote:
> ObUL: You know it's going to be cited by people as an example
> of those crazy scientists wasting public money.
Or at least public monkey.
Charles "Our future seems to look pretty funky" Lieberman
--
Charles A. Lieberman | "Granted, the animals without heads, bones, or
Brooklyn, NY, USA | limbs need a lot of assistance to breed, but so
cali...@bigfoot.com | what?" Nathan Tenny teaches AFU animal husbandry
Lon Stowell
> In article <3ebbef93....@news.cpqcorp.net>,
> jeff.lana...@compaq-dot-com.not (Jeff Lanam) wrote:
>
>
>>ObUL: You know it's going to be cited by people as an example
>>of those crazy scientists wasting public money.
>
>
> Or at least public monkey.
Do the monkeys practice on scratch paper, or is that only
the scratch monkey?
Dr H
On Fri, 9 May 2003, John Schmitt wrote:
}Today's Times contains a report about an attempt to demonstrate
}the monkeys-typewriters-Shakespeare theory by experiment. If you
}go to their homepage and search using the obvious terms in the
}near future, you can read an article which will, at the very
}least, make the corners of you mouth turn upwards. It is also at
}the BBC site:
}
}http://news.bbc.co.uk/1/hi/england/devon/3013959.stm
}
}Admittedly it is a very small sample, with hardly enough monkeys,
}typewriters or time, but it is heartening to hear that bashing a
}horse until only a faint stain remains is still a sport academia
}supports.
"Paignton Zoo scientific officer Dr Amy Plowman said: "The work was
interesting but had little scientific value, except to show that
the 'infinite monkey' theory is flawed.""
Huh? It showed nothing of the sort. The whole experiment was flawed.
They didn't have nearly enough monkeys, it didn't go on for nearly long
enough, and they were supposed to use *typewriters*, not a *computer*.
Sheesh.
Dr H
TeaLady (Mari C.)
news:Pine.GSU.4.21.030512...@garcia.efn.org:
> }Admittedly it is a very small sample, with hardly enough
> monkeys, }typewriters or time, but it is heartening to hear
> that bashing a }horse until only a faint stain remains is still
> a sport academia }supports.
>
>
> "Paignton Zoo scientific officer Dr Amy Plowman said: "The
> work was interesting but had little scientific value, except
> to show that the 'infinite monkey' theory is flawed.""
>
> Huh? It showed nothing of the sort. The whole experiment was
> flawed. They didn't have nearly enough monkeys, it didn't go on
> for nearly long enough, and they were supposed to use
> *typewriters*, not a *computer*.
>
> Sheesh.
>
Certainly sheesh. The damnable beasts probably hooked into the
internet and spent the whole time trying to download porn. No
wonder they shat upon the keyboards - there isn't a decent hot
monkey sex site out there, at least not one for *real* monkeys.
--
TeaLady (mari)
...But now I'm feeling so much better, I could cakewalk into town.
R H Draney
>> work was interesting but had little scientific value, except
> > to show that the 'infinite monkey' theory is flawed.""
>>
>> Huh? It showed nothing of the sort. The whole experiment was
>> flawed. They didn't have nearly enough monkeys, it didn't go on
>> for nearly long enough, and they were supposed to use
>> *typewriters*, not a *computer*.
Well, they did appear to establish a result that calls into question the
original theory: the output of the group of monkeys consisted largely of several
pages of the same letter...the assumption before this was that they'd produce
all the letters more or less at random....
>Certainly sheesh. The damnable beasts probably hooked into the
>internet and spent the whole time trying to download porn. No
>wonder they shat upon the keyboards - there isn't a decent hot
>monkey sex site out there, at least not one for *real* monkeys.
Cite?
....r
--
"Bill sings to Sarah. Sarah sings to Bill. Perhaps they will do other
dangerous things together. They may eat lamb or stroke each other.
They may chant of their difficulties and their happiness. They have
love but they also have typewriters. That is interesting." - Racter
Dan Fingerman
<news:b9peu...@drn.newsguy.com>:
> In article <Xns9379D18...@130.133.1.4>, "TeaLady says...
>>
>>Dr H <hiaw...@efn.org> wrote in
>>news:Pine.GSU.4.21.030512...@garcia.efn.org:
>>
>>> "Paignton Zoo scientific officer Dr Amy Plowman said: "The
>>> work was interesting but had little scientific value, except
>> > to show that the 'infinite monkey' theory is flawed.""
>>>
>>> Huh? It showed nothing of the sort. The whole experiment was
>>> flawed. They didn't have nearly enough monkeys, it didn't go on
>>> for nearly long enough, and they were supposed to use
>>> *typewriters*, not a *computer*.
>
> Well, they did appear to establish a result that calls into
> question the original theory: the output of the group of monkeys
> consisted largely of several pages of the same letter...the
> assumption before this was that they'd produce all the letters
> more or less at random....
Yeah, but: "But towards the end of the experiment, their output
slightly improved, with the letters A, J, L and M also appearing."
Sounds like the zookeepers ran out of time/funding/patience just as
the monkeys were shedding their writer's block.
--
DTM :<|
R H Draney
>> question the original theory: the output of the group of monkeys
>> consisted largely of several pages of the same letter...the
>> assumption before this was that they'd produce all the letters
>> more or less at random....
>
>Yeah, but: "But towards the end of the experiment, their output
>slightly improved, with the letters A, J, L and M also appearing."
>Sounds like the zookeepers ran out of time/funding/patience just as
>the monkeys were shedding their writer's block.
Or, sensing the impending end of the experiment, the monkeys demonstrated a
simian Hawthorne effect?...r
Donna Richoux
> In article <Xns9379D18...@130.133.1.4>, "TeaLady says...
> >
> >Dr H <hiaw...@efn.org> wrote in
> >news:Pine.GSU.4.21.030512...@garcia.efn.org:
> >
> >> "Paignton Zoo scientific officer Dr Amy Plowman said: "The
> >> work was interesting but had little scientific value, except
> > > to show that the 'infinite monkey' theory is flawed.""
> >>
> >> Huh? It showed nothing of the sort. The whole experiment was
> >> flawed. They didn't have nearly enough monkeys, it didn't go on
> >> for nearly long enough, and they were supposed to use
> >> *typewriters*, not a *computer*.
>
> Well, they did appear to establish a result that calls into question the
> original theory: the output of the group of monkeys consisted largely of
> several pages of the same letter...the assumption before this was that
> they'd produce all the letters more or less at random....
We keep trying to tell y'all that flipping a fair coin *can* produce a
very long string of heads, but do you believe us? No, you have these
preconceived notions of what "random" looks like...
--
Donna "see Rosencrantz & Guildenstern Are Dead, Act I" Richoux
Harold Buck
tr...@euronet.nl (Donna Richoux) wrote:
What about when they typed "It was the best of times, it was the blurst
of times"?
--Harold Buck
"I used to rock and roll all night,
and party every day.
Then it was every other day. . . ."
-Homer J. Simpson
Lee Ayrton
> ObUL: You know it's going to be cited by people as an example
> of those crazy scientists wasting public money.
It has started already. Just yesterday I heard a lo cal disk jockey
smugly blathering about wasted research money and the typing monkeys.
Lee "The study did demonstrate that it wouldn't take too many monkeys to
reproduce JD on-air patter" Ayrton
--
"I tend to state my position in confident declarative sentences, but the
possibility that I'm a wrong-minded dipshit asshole is never far from my
mind." Andy Walton teaches ballroom dancing in AFU.
Joseph Michael Bay
>Certainly sheesh. The damnable beasts probably hooked into the
>internet and spent the whole time trying to download porn. No
>wonder they shat upon the keyboards - there isn't a decent hot
>monkey sex site out there, at least not one for *real* monkeys.
I heard that several of the monkeys stopped eating for several
days after misreading a spam in their inbox as "MAKE MONKEYS FAST".
--
Joseph M. Bay Lamont Sanford Junior University
www.stanford.edu/~jmbay/
n o m a t t e r h o w r o u n d i t f e e l s
i t ' s s t i l l f l a t
Joseph Michael Bay
>Should David Ives' play, "All in the Timing" be performed
>anywhere in the vicinity I recommend everyone to see it. One of
>the scenes involves severaal monkeys which have been put to the
>task of writing Shakespeare. Another scene involves a number of
>variations on how Leon Trotsky came to have a pick stuck in his
>head.
GAH! DAMMIT! DAMN DAMN DAMN!
MOUNTAIN CLIMBER'S AXE!
Joe "can't I get that through your skull?" Bay
R H Draney
>
>It has started already. Just yesterday I heard a lo cal disk jockey
>smugly blathering about wasted research money and the typing monkeys.
>
>Lee "The study did demonstrate that it wouldn't take too many monkeys to
>reproduce JD on-air patter" Ayrton
I never noticed before, but there *do* seem to be a lot of juvenile delinquents
on radio these days....r
R H Draney
says...
>
>What about when they typed "It was the best of times, it was the blurst
>of times"?
Even with the obvious spelling correction, it's still nonsense...how can it be
the best *and* the worst of times?...that's like saying "the sky is green, the
sky is red"....
Silly monkeys....
Just had an interesting thought...suppose you took the monkeys' output and fed
it through an automatical spelling corrector before letting anyone see
it...would *that* increase the chances that they'd come up with something
useful?...r
Lee Ayrton
> >It has started already. Just yesterday I heard a lo cal disk jockey
> >smugly blathering about wasted research money and the typing monkeys.
> >
> >Lee "The study did demonstrate that it wouldn't take too many monkeys to
> >reproduce JD on-air patter" Ayrton
>
> I never noticed before, but there *do* seem to be a lot of juvenile delinquents
> on radio these days....r
Oh. Ow. Argh. Did I really tpye that?
Lee "Excuse me, I just remembered an appointment. Must go. Bye!" Ayrton
Lon Stowell
Are you asking for commentary on the output of monkeys,
or seeking discussion on the bane of the universe and
destroyer of mentality, the spellchecker.
Lon Stowell
> "TeaLady (Mari C.)" <spres...@yahoo.com> writes:
>
>
>
>>Certainly sheesh. The damnable beasts probably hooked into the
>>internet and spent the whole time trying to download porn. No
>>wonder they shat upon the keyboards - there isn't a decent hot
>>monkey sex site out there, at least not one for *real* monkeys.
>
>
> I heard that several of the monkeys stopped eating for several
> days after misreading a spam in their inbox as "MAKE MONKEYS FAST".
You do realize that someday, somehow, somewhere,
you will be forced to pay for that remark.
Larry D. Farrell
R H Draney wrote:
> Just had an interesting thought...suppose you took the monkeys' output and fed
> it through an automatical spelling corrector before letting anyone see
> it...would *that* increase the chances that they'd come up with something
> useful?...r
If you put these monkeys' output through a spelling corrector, wouldn't it still
be shit?
Richard Muth
> In article <no_one_knows-E274...@netnews.attbi.com>, Harold
> says...
> >
> >What about when they typed "It was the best of times, it was the blurst
> >of times"?
>
> Even with the obvious spelling correction, it's still nonsense...how can it be
> the best *and* the worst of times?...that's like saying "the sky is green, the
> sky is red"....
No, no, no. They actually typed "It was the best of times, it was the wurst of
times." Obviously trying to start another food thread...
--
Richard F. Muth, MS, RN
Environmental, Health & Safety Officer
Gordon College, Wenham, MA 01984-1899
Lon Stowell
No, it is very unlikely that "output" would be rendered thusly.
You'd probably have to start with a monkey'd "shet" or similar.
Dr H
On 12 May 2003, R H Draney wrote:
}In article <Xns9379D18...@130.133.1.4>, "TeaLady says...
}>
}>Dr H <hiaw...@efn.org> wrote in
}>news:Pine.GSU.4.21.030512...@garcia.efn.org:
}>
}>> "Paignton Zoo scientific officer Dr Amy Plowman said: "The
}>> work was interesting but had little scientific value, except
}> > to show that the 'infinite monkey' theory is flawed.""
}>>
}>> Huh? It showed nothing of the sort. The whole experiment was
}>> flawed. They didn't have nearly enough monkeys, it didn't go on
}>> for nearly long enough, and they were supposed to use
}>> *typewriters*, not a *computer*.
}
}Well, they did appear to establish a result that calls into question the
}original theory: the output of the group of monkeys consisted largely of several
}pages of the same letter...the assumption before this was that they'd produce
}all the letters more or less at random....
Random sequences can, and frequently do include long sections of a
single value. The sequences are no less random for the fact that
we tend to notice groups of similar items and /assume/ they represent
a deliberate pattern.
Dr H
R H Draney
>
> Random sequences can, and frequently do include long sections of a
> single value. The sequences are no less random for the fact that
> we tend to notice groups of similar items and /assume/ they represent
> a deliberate pattern.
I see it more as evidence that monkeys are, like human beings, incapable of
behaving in a truly random fashion...I thought the quote was a little pithier
(and said something about human beings being unable to create a random
sequence), but from Knuth's "The Art of Computer Programming: Volume 2,
Seminumerical Algorithms (3rd edition)", it takes the form of the first exercise
in the section on random-number generators:
1. [20] Suppose that you wish to obtain a decimal digit at random, not using a
computer. Which of the following methods would be suitable?
...
f) Ask a friend to think of a random digit, and use the digit he names.
g) Ask an enemy to think of a random digit, and use the digit he names.
...
From the Answers section:
...(f,g) No. People usually think of certain digits (like 7) with higher
probability....
To which I might add: "monkeys usually hit certain keys on a typewriter (like S)
with higher probability"...we now have empirical evidence for the latter....r
Old Hippy Bastard
> Joseph Michael Bay wrote:
>
>> "TeaLady (Mari C.)" <spres...@yahoo.com> writes:
>>
>>
>> I heard that several of the monkeys stopped eating for several
>> days after misreading a spam in their inbox as "MAKE MONKEYS FAST".
>
>
> You do realize that someday, somehow, somewhere,
> you will be forced to pay for that remark.
>
>
Damn, that makes the third keyboard I've ruined this month
by spitting beer into it while relaxing by reading AFU and
such.
O.H.B. (internymically challenged)
Crashj
<>
> I heard that several of the monkeys stopped eating for several
> days after misreading a spam in their inbox as "MAKE MONKEYS FAST".
No, it was the "Grow Bigger Peanuts" that got to them.
Crashj 'satisfied with mine' Johnson
Nathan Tenny
(1) I know you're being whimsical; and, in consequence,
(2) I know I'm sort of being a humorless dork here. Live with it.
Of course, the *likelihood* of a random process producing a string of
heads of a particular length can be known; and he did say "calls into
question", not "disproves".
Goes to the subtle difference between "this sequence is random" (which
doesn't mean anything) and "this sequence was produced by a random
process" (which can be studied as to its probability). For lo, on such
stuff is statistical significance made.
NT
--
Nathan Tenny | When the world ends, there'll be no more
Qualcomm, Inc., San Diego, CA | air. That's why it's important to pollute
<nten...@qualcomm.com> | the air now. Before it's too late.
| -- Kathy Acker
John Schmitt
cra...@mindspring.com (Crashj) writes:
>> I heard that several of the monkeys stopped eating for several
>> days after misreading a spam in their inbox as "MAKE MONKEYS FAST".
>No, it was the "Grow Bigger Peanuts" that got to them.
Bearing in mind their text output, I suggest that their
unfamiliarity with the keyboard caused them to misinterpret
"Make $$$$$$$".
John "sometimes spam makes me want to crap on the keyboard, too"
Schmitt
--
If you have nothing to say, or rather, something extremely stupid
and obvious, say it, but in a 'plonking' tone of voice - i.e.
roundly, but hollowly and dogmatically. - Stephen Potter
Charles A Lieberman
n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m (Nathan Tenny) wrote:
> Of course, the *likelihood* of a random process producing a string of
> heads of a particular length can be known
Yes, the probability that n coin flips will result in n heads is 2^-n.
Charles "same as in town" Lieberman
--
Charles A. Lieberman | "Granted, the animals without heads, bones, or
Brooklyn, NY, USA | limbs need a lot of assistance to breed, but so
cali...@bigfoot.com | what?" Nathan Tenny teaches AFU animal husbandry
Crashj
<>
> No, you have these
> preconceived notions of what "random" looks like...
"A wily-looking little man, with a sharp nose and a laughing mouth and
a shock of straw-colored hair. He was dressed in something like a
Renaissance costume of orange, red, and brown. He wore long hose and a
tight-fitting embroidered doublet. His name was Random."
Crashj 'Rog, you there?' Johnson
Lon Stowell
Like the other 8 princes, he may be stuck in something
very old and formerly sticky.
Harold Buck
Dr H <hiaw...@efn.org> wrote:
>
> Random sequences can, and frequently do include long sections of a
> single value. The sequences are no less random for the fact that
> we tend to notice groups of similar items and /assume/ they represent
> a deliberate pattern.
>
And such coincidences are the basis of much of the "paranormal": people
are very good at finding patterns in random data, and then they feel the
need to "explain" it.
Harold Buck
Lee Ayrton <lay...@ntplx.net> wrote:
>
> In article <3ebbef93....@news.cpqcorp.net>,
> jeff.lana...@compaq-dot-com.not (Jeff Lanam) wrote:
>
> > ObUL: You know it's going to be cited by people as an example
> > of those crazy scientists wasting public money.
>
> It has started already. Just yesterday I heard a lo cal disk jockey
> smugly blathering about wasted research money and the typing monkeys.
>
Mmmmmmm. . . .Lo cal disck jockey. . . .Tastes great. . . . Less
filling. . . .
Hatunen
<no_one...@attbi.com> wrote:
>In article <Pine.GSU.4.21.03051...@garcia.efn.org>,
> Dr H <hiaw...@efn.org> wrote:
>
>>
>> Random sequences can, and frequently do include long sections of a
>> single value. The sequences are no less random for the fact that
>> we tend to notice groups of similar items and /assume/ they represent
>> a deliberate pattern.
>>
>And such coincidences are the basis of much of the "paranormal": people
>are very good at finding patterns in random data, and then they feel the
>need to "explain" it.
An infinitely long string of random digits will have an
infinitely long string of 2s in it.
************* DAVE HATUNEN (hat...@cox.net) *************
* Tucson Arizona, out where the cacti grow *
* My typos & mispellings are intentional copyright traps *
Harold Buck
hatu...@cox.net (Hatunen) wrote:
>
> An infinitely long string of random digits will have an
> infinitely long string of 2s in it.
It's unclear if you're joking, but I hope so. There is no upper bound to
the length of a run of 2's in the sequence, but the only way you can
have an infinitely long sequence of 2's is if it ends with a string of
repeating 2's. Thus, you'd be saying that every sequence of random
digits ends in a string of 2's repeating forever.
Of course, since the same thing would be true of *every* digit, which
creates some problems.
Oh, and there are issues with the way you've stated things. If I flip a
coin repeatedly and write down a 1 for heads and a 0 for tails, I'll get
an infinitely long string of random digits. Of course, NONE of them will
be 2's.
John Francis
Harold Buck <no_one...@attbi.com> wrote:
>In article <3ec2e4ae...@news.west.cox.net>,
> hatu...@cox.net (Hatunen) wrote:
>
>>
>> An infinitely long string of random digits will have an
>> infinitely long string of 2s in it.
>
>
>It's unclear if you're joking, but I hope so. There is no upper bound to
>the length of a run of 2's in the sequence, but the only way you can
>have an infinitely long sequence of 2's is if it ends with a string of
>repeating 2's.
Nope. Back to school for you. You can have that infinitely long string
of twos, followed by an infinitely long string of threes, followed by an
infinitely long string of fours, ...
--
As evil plans go, it doesn't suck -- Wesley offers a critique on "Angel"
Brett Buck
There is no upper bound to
> the length of a run of 2's in the sequence, but the only way you can
> have an infinitely long sequence of 2's is if it ends with a string of
> repeating 2's.
Ends?
Brett "no relation as far as I know" Buck
Crashj
I still tell the wife I am taking her out to Kenny Roi's Fried Lizard.
Crashj 'now known as KFL' Johnson
Nathan Tenny
Hatunen <hatu...@cox.net> wrote:
>An infinitely long string of random digits will have an
>infinitely long string of 2s in it.
I can't decide if this is right or not. It certainly is if you add "with
probability 1" to the end, but that isn't quite the same as "absolutely".
There are (of course) many infinite sequences of digits that don't contain
an infinitely long string of 2s. What I can't decide is whether it's
possible or not for a random process to produce one of them. It's very
*unlikely*, to be sure, but then any *particular* sequence is equally
unlikely...and for that reasons, isn't an endless string of 1s (or any
other sequence chosen not to have an infinite string of 2s) no *more*
unlikely than any other output?
I'm just a humble low-d topologist. Big numbers make my brane hert.
Hatunen
(Nathan Tenny) wrote:
>In article <3ec2e4ae...@news.west.cox.net>,
>Hatunen <hatu...@cox.net> wrote:
>>An infinitely long string of random digits will have an
>>infinitely long string of 2s in it.
>
>I can't decide if this is right or not. It certainly is if you add "with
>probability 1" to the end, but that isn't quite the same as "absolutely".
Then how do you define a probability of 1?
>There are (of course) many infinite sequences of digits that don't contain
>an infinitely long string of 2s.
Yep. An infinite number of them. But that doesn't preclude also
having an infinite run of 2s. And 3s.
>What I can't decide is whether it's
>possible or not for a random process to produce one of them. It's very
>*unlikely*, to be sure, but then any *particular* sequence is equally
>unlikely...
And equally likely. In fact, every sequence you can think of will
be in there.
>and for that reasons, isn't an endless string of 1s (or any
>other sequence chosen not to have an infinite string of 2s) no *more*
>unlikely than any other output?
Yep. I just used one example.
Nathan Tenny
Hatunen <hatu...@cox.net> wrote:
>On 14 May 2003 20:31:25 -0700, n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m
>(Nathan Tenny) wrote:
>>In article <3ec2e4ae...@news.west.cox.net>,
>>Hatunen <hatu...@cox.net> wrote:
>>>An infinitely long string of random digits will have an
>>>infinitely long string of 2s in it.
>>
>>I can't decide if this is right or not. It certainly is if you add "with
>>probability 1" to the end, but that isn't quite the same as "absolutely".
>
>Then how do you define a probability of 1?
In terms of measures, which in the Land of the Infinite basically means
as a limit, with the attendant confusion about whether "x approaches 1"
really does mean "in the infinite view x is 1".
["them"=="infinite sequences without an infinite run of 2s"]
>>What I can't decide is whether it's
>>possible or not for a random process to produce one of them. It's very
>>*unlikely*, to be sure, but then any *particular* sequence is equally
>>unlikely...
>
>And equally likely. In fact, every sequence you can think of will
>be in there.
Isn't that the opposite of what you said at the beginning? If "every
sequence you can think of is in there" (and I agree with that), then
all the infinitely many sequences *without* an infinite run of 2s are
"in there", where "in there" means "possible output of a random process".
Now I've confused myself even more by mentioning measures. Lemme see.
We agree for sure on the statement
An infinite sequence of integers generated by a random process
contains an infinite subsequence of 2s, with probability 1.
Really, though, by "infinite sequence of integers" we mean "arbitrary
mapping Z+ -> Z". I'm unsure how to characterize "contains an infinite
subsequence of 2s", but I don't think it's very important---I'm pretty
sure we agree on everything salient about that property, so lemme just
call it Property S. The statement above is synonymous with
A randomly chosen mapping from Z+ to Z has property S, with probability 1.
What *that* means is
The set of mappings Z+ -> Z *without* property S has measure zero
in the space of all mappings Z+ -> Z.
I'm a little nervous about just what the measure on a space of mappings
with discrete codomains should be for this purpose, though; but let it
stand.
Point is, you pick a random one of those mappings, and you *may* get
one of the non-S ones---the set of them has measure zero, but that
doesn't mean you *never* hit one, just that as you keep throwing darts
at the mapping space, the fraction of times that you hit a non-S
target approaches zero as the number of trials approaches infinity.
And that's the difference between "absolutely" and "with probability 1".
I'd be happier with this discussion if someone could remind me what the
appropriate measure is. I barely made it through my first-year real-
analysis course, and that was thirteen years ago.
Aaron Davies
> In article <3ec2e4ae...@news.west.cox.net>,
> Hatunen <hatu...@cox.net> wrote:
> >An infinitely long string of random digits will have an
> >infinitely long string of 2s in it.
>
> I can't decide if this is right or not. It certainly is if you add "with
> probability 1" to the end, but that isn't quite the same as "absolutely".
>
> There are (of course) many infinite sequences of digits that don't contain
> an infinitely long string of 2s. What I can't decide is whether it's
> possible or not for a random process to produce one of them. It's very
> *unlikely*, to be sure, but then any *particular* sequence is equally
> unlikely...and for that reasons, isn't an endless string of 1s (or any
> other sequence chosen not to have an infinite string of 2s) no *more*
> unlikely than any other output?
Um, isn't a random process that includes 2s among the digits it
generates guaranteed to generate an infinite string of 2s if it runs for
an infinite amount of time? I would think all possible outputs would
have to be generated given an infinite amount of time; or to put it
another way, if there's a finite probability of an event's occuring,
then given an infinite amount of time, the probability of its *not*
occuring should be zero. Or maybe not. IANAStatistician.
--
__ __
/ ) / )
/--/ __. __ ________ / / __. , __o _ _
/ (_(_/|_/ (_(_) / / <_ /__/_(_/|_\/ <__</_/_)_
Harold Buck
n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m (Nathan Tenny) wrote:
> In article <3ec2e4ae...@news.west.cox.net>,
> Hatunen <hatu...@cox.net> wrote:
> >An infinitely long string of random digits will have an
> >infinitely long string of 2s in it.
>
> I can't decide if this is right or not. It certainly is if you add "with
> probability 1" to the end, but that isn't quite the same as "absolutely".
>
> There are (of course) many infinite sequences of digits that don't contain
> an infinitely long string of 2s. What I can't decide is whether it's
> possible or not for a random process to produce one of them. It's very
> *unlikely*, to be sure, but then any *particular* sequence is equally
> unlikely...and for that reasons, isn't an endless string of 1s (or any
> other sequence chosen not to have an infinite string of 2s) no *more*
> unlikely than any other output?
The probability of any particular infinite string is 0.
Harold Buck
jo...@panix.com (John Francis) wrote:
Um, where did you get YOUR masters degree in math? For it to be
infinite, there's only one direction it can expand, and that's to the
right. Now, the length of the longest run of 2's, 3's, and so on can
each be UNBOUNDED, which is not the same as being infinitely long.
Consider:
123112233111222333.... (pattern repeats)
The length of the strings of 1's, 2's, and 3's are unbounded, but there
are no infinitely long strings of any of them. Of course, this isn't a
random string, but the same logic applies.
Now, if you want an infinitely long string, it would need something like:
2379872458972359877222222222222222... (2's repeat)
If the tail end doesn't end with the same number repeating, it isn't an
inifinite string. Period.
Binyamin Dissen
[ snipped ]
:>Now, if you want an infinitely long string, it would need something like:
:>2379872458972359877222222222222222... (2's repeat)
:>If the tail end doesn't end with the same number repeating, it isn't an
:>inifinite string. Period.
The set of even integers is infinite.
The set of all integers are infinite.
An infinite set can contain other infinite sets. Period.
[ snipped ]
--
Binyamin Dissen <bdi...@dissensoftware.com>
http://www.dissensoftware.com
Charles A Lieberman
Harold Buck <no_one...@attbi.com> wrote:
> > An infinitely long string of random digits will have an
> > infinitely long string of 2s in it.
>
>
> It's unclear if you're joking, but I hope so. There is no upper bound to
> the length of a run of 2's in the sequence, but the only way you can
> have an infinitely long sequence of 2's is if it ends with a string of
> repeating 2's. Thus, you'd be saying that every sequence of random
> digits ends in a string of 2's repeating forever.
No, he's saying one such sequence ends thus. Actually, there are an
infinite number of sequences that do, but these are not all of the
infinite number of infinite sequences.
Harold Buck
Binyamin Dissen <post...@dissensoftware.com> wrote:
> On Thu, 15 May 2003 12:19:33 GMT Harold Buck <no_one...@attbi.com> wrote:
>
> [ snipped ]
>
> :>Now, if you want an infinitely long string, it would need something like:
>
> :>2379872458972359877222222222222222... (2's repeat)
>
> :>If the tail end doesn't end with the same number repeating, it isn't an
> :>inifinite string. Period.
>
> The set of even integers is infinite.
>
> The set of all integers are infinite.
>
> An infinite set can contain other infinite sets.
Yeah, I know that. I apparently know a lot more about it than you,
though.
An infinitely long string of 2's cannot contain an infinitely long
string of 3's, or any 3's at all for that matter, because the
implication of the term "an infinitely long string of 2's" is the the
2's are CONSECUTIVE.
Certainly any string of digits in which each number appears with
probability 1/10 contains infinitely many 0's, 1's, 2's, ... , and 9's,
but that is most assuredly NOT the same as having infinitely long
STRINGS of these digits.
Again, there length of such strings are UNBOUNDED (no maximum length),
but that is NOT the same as being infinitely long.
If you still disagree, perhaps you can explain what you think a sequence
looks like that has an infinite string of 2's and also an infinite
string of 3's. If you can't do it, you'll have to give up on your
argument. If you can explain what you mean, it will be easier for me to
point out where you're going wrong.
Harold Buck
Charles A Lieberman <cali...@bigfoot.com> wrote:
> In article <no_one_knows-4F18...@netnews.attbi.com>,
> Harold Buck <no_one...@attbi.com> wrote:
>
> > > An infinitely long string of random digits will have an
> > > infinitely long string of 2s in it.
> >
> >
> > It's unclear if you're joking, but I hope so. There is no upper bound to
> > the length of a run of 2's in the sequence, but the only way you can
> > have an infinitely long sequence of 2's is if it ends with a string of
> > repeating 2's. Thus, you'd be saying that every sequence of random
> > digits ends in a string of 2's repeating forever.
>
> No, he's saying one such sequence ends thus. Actually, there are an
> infinite number of sequences that do, but these are not all of the
> infinite number of infinite sequences.
No, that's not what he's saying. I read it to say "[If you have] An
infinitely long string of random digits [it] will have an infinitely
long string of 2s in it."
And that's just rubbish.
And if you have an infinitely long string of random digits (each digit
with probability 1/10, say), the probability that it has an infinitely
long string of any digit is 0.
However, there is NO UPPER BOUND to the length of a string of 2's.
That's not the same as being infinitely long.
Binyamin Dissen
:>In article <r037cvchb209vsb6o...@4ax.com>,
:> Binyamin Dissen <post...@dissensoftware.com> wrote:
:>> On Thu, 15 May 2003 12:19:33 GMT Harold Buck <no_one...@attbi.com> wrote:
:>> [ snipped ]
:>> :>Now, if you want an infinitely long string, it would need something like:
:>> :>2379872458972359877222222222222222... (2's repeat)
:>> :>If the tail end doesn't end with the same number repeating, it isn't an
:>> :>inifinite string. Period.
:>> The set of even integers is infinite.
:>> The set of all integers are infinite.
:>> An infinite set can contain other infinite sets.
:>Yeah, I know that. I apparently know a lot more about it than you,
:>though.
Interesting assertion.
:>An infinitely long string of 2's cannot contain an infinitely long
:>string of 3's, or any 3's at all for that matter, because the
:>implication of the term "an infinitely long string of 2's" is the the
:>2's are CONSECUTIVE.
No. A group of only foo's will not have any bar's if they are different sets.
An infinitely long string of digits can contain an infinite numbers of
infinite strings containing the digit 2 alone.
:>Certainly any string of digits in which each number appears with
:>probability 1/10 contains infinitely many 0's, 1's, 2's, ... , and 9's,
:>but that is most assuredly NOT the same as having infinitely long
:>STRINGS of these digits.
As the string is infinitely long, it will contain an infinite substring with
the digit 2 alone.
:>Again, there length of such strings are UNBOUNDED (no maximum length),
:>but that is NOT the same as being infinitely long.
Irrelevant.
:>If you still disagree, perhaps you can explain what you think a sequence
:>looks like that has an infinite string of 2's and also an infinite
:>string of 3's. If you can't do it, you'll have to give up on your
:>argument. If you can explain what you mean, it will be easier for me to
:>point out where you're going wrong.
Simple.
An infinitely long string consisting of an infinite number of substrings, many
being substrings of infinite length containing the digit 2 alone, and many
containing the digit 3 alone.
Your misunderstanding is that you do not understand the concept of infinity.
Barbara Needham
> An infinitely long string consisting of an infinite number of substrings, many
> being substrings of infinite length containing the digit 2 alone, and many
> containing the digit 3 alone.
>
> Your misunderstanding is that you do not understand the concept of infinity.
Well, I probably know less than anybody. But back in the dark ages when
I took math, were there not two [or more?] kinds of infinity? At least I
remember the kind that goes on and on and on such as the series of
counting numbers to the highest number, and then the kind where you can
divide an inch into in infinite number of "pieces", but that one
wouldn't work with integers... And something seemed wrong the with the
concept of 2's at the "end" because infinity doesn't have an end does
it? Only a limit? In some cases...
Clarification, Binyamin, please...
--
Barbara Needham
John Francis
Harold Buck <no_one...@attbi.com> wrote:
>In article <b9v0ds$hlh$1...@panix5.panix.com>,
> jo...@panix.com (John Francis) wrote:
>
>> In article <no_one_knows-4F18...@netnews.attbi.com>,
>> Harold Buck <no_one...@attbi.com> wrote:
>> >In article <3ec2e4ae...@news.west.cox.net>,
>> > hatu...@cox.net (Hatunen) wrote:
>> >
>> >>
>> >> An infinitely long string of random digits will have an
>> >> infinitely long string of 2s in it.
>> >
>> >
>> >It's unclear if you're joking, but I hope so. There is no upper bound to
>> >the length of a run of 2's in the sequence, but the only way you can
>> >have an infinitely long sequence of 2's is if it ends with a string of
>> >repeating 2's.
>>
>> Nope. Back to school for you. You can have that infinitely long string
>> of twos, followed by an infinitely long string of threes, followed by an
>> infinitely long string of fours, ...
>
>
>Um, where did you get YOUR masters degree in math?
I'll tell you, when and if I think it's important.
> For it to be
>infinite, there's only one direction it can expand, and that's to the
>right.
That's an assertion. Prove it.
[ . . . . ]
>Now, if you want an infinitely long string, it would need something like:
>
>2379872458972359877222222222222222... (2's repeat)
Another assertion, and a false one at that.
>If the tail end doesn't end with the same number repeating, it isn't an
>inifinite string. Period.
So if you take an infinite string of digits which starts of as all 2s,
then switches to 3s 'half way', how many 2s are there?
You might want to read some elementary introduction to infinities before
you dig yourself a much deeper hole. Conway's "On numbers and games"
is one such which is accessible but still sufficiently rigorous.
Conway doesn't seem to have a problem with infinite sequences included
as interior subsequences of another sequence, and I'm prepared to wager
that he's a better mathematician than either of us.
Lee Rudolph
or the guilty, but just because it's too damned hard to untangle
this mess.
>>> >> An infinitely long string of random digits will have an
>>> >> infinitely long string of 2s in it.
In this assertion, the term "string" appears; I believe it has
not been defined in the present thread. The term "in it" also
appears without definition.
>>> >It's unclear if you're joking, but I hope so. There is no upper bound to
>>> >the length of a run of 2's in the sequence, but the only way you can
>>> >have an infinitely long sequence of 2's is if it ends with a string of
>>> >repeating 2's.
In *this* assertion, the terms "run" and "sequence" appear, along
with "string"; nor have they been defined.
>>> Nope. Back to school for you. You can have that infinitely long string
>>> of twos, followed by an infinitely long string of threes, followed by an
>>> infinitely long string of fours, ...
In this assertion, it is apparently assumed that an "infinitely
long string" (not defined) can be "followed by" something (in
particular, by other strings), and it is furthermore apparently
assumed (else there would be little point in making the assertion)
that an "infinitely long string" "followed by an infinitely long
string" is still a "string".
>You might want to read some elementary introduction to infinities before
>you dig yourself a much deeper hole. Conway's "On numbers and games"
>is one such which is accessible but still sufficiently rigorous.
>Conway doesn't seem to have a problem with infinite sequences included
>as interior subsequences of another sequence, and I'm prepared to wager
>that he's a better mathematician than either of us.
And, finally, these statements include an assertion refering to
"sequences" again, as well as "interior subsequences" thereof,
but do not define either of those terms (though they suggest--
my guess is, slightly abusively [in the technical sense, and I
don't mean Lord Romial's technical sense, I mean Bourbaki's]--
that ONAG defines "sequences" purely and simply).
Now, we can all define our terms (in particular the terms "string",
"sequence", etc.) as we wish, if we make our definitions clear; but
in the absence of explicit definitions, I think it's reasonable to
assume the standard definitions used in the field, when there are
such (and when there aren't, or when there are competing standards,
then we run the risk of pointless arguments). As *I* understand
it, a "string" (unless otherwise noted) is a finite or denumerably
infinite sequence, whose order type is either one of the finite
the ordinals 0, 1, 2, ..., or the ordinal \omega which is the
order type of the positive integers. In particular, as *I*
understand it, while there is an operation of "concatenation"
of *finite* strings, and an operation by which an infinite
string can be concatenated to a finite string, both such
operations yielding strings, it is *not* possible to concatenate
two infinite strings and get a string as the result. You get
an ordered set of order type \omega + \omega, which is not equal
to \omega (as an ordinal). On that understanding, you cannot
have "an infinitely long string of twos, followed by an
infinitely long string of threes" and fairly say that what you
have is itself a "string".
...This is rapidly exhausting my pendantry. Let me just repeat
explicitly my implicit point above: with so many undefined terms
floating around, it's very hard to tell who's saying what, much
less to say whether the various statements are (a) meaningful,
(b) true, (c) in contradiction to each other, or (d) all and none
of the above.
Oh, and I never got a master's degree. But I've had lunch with
Conway (once) and exchanged several e-mails with him (he sent me
I was an ASCII diagram of a knot he discovered that I was interested in).
Lee "Ph.D. (MIT), M-O-U-S-E!" Rudolph
Harold Buck
jo...@panix.com (John Francis) wrote:
>
> >Now, if you want an infinitely long string, it would need something like:
> >
> >2379872458972359877222222222222222... (2's repeat)
>
> Another assertion, and a false one at that.
>
> >If the tail end doesn't end with the same number repeating, it isn't an
> >inifinite string. Period.
>
> So if you take an infinite string of digits which starts of as all 2s,
> then switches to 3s 'half way', how many 2s are there?
Okay, the only thing I can see here is that your definition of "an
infinite string" differs from mine. Just to check, tell me this:
If the "string" of digits is formed by choosing a digit from 0, 1, ...,
9 with equal probability, and writing it down, and then repeating over
and over again, do you still claim that you can have an infinite
"string" of 2's and also an infinite "string" of 3's in the same
"string"?
This model seems to be the most appropriate when we're talking about a
monkey banging on a keyboard.
Charles A Lieberman
Harold Buck <no_one...@attbi.com> wrote:
> > No, he's saying one such sequence ends thus. Actually, there are an
> > infinite number of sequences that do, but these are not all of the
> > infinite number of infinite sequences.
>
>
> No, that's not what he's saying. I read it to say "[If you have] An
> infinitely long string of random digits [it] will have an infinitely
> long string of 2s in it."
Well, that appears to be the problem. You're wrong. He certainly didn't
say "*Every* infinitely long string of random digits will have an
infinitely long string of 2s in it."
--
Harold Buck
Charles A Lieberman <cali...@bigfoot.com> wrote:
> In article <no_one_knows-5948...@netnews.attbi.com>,
> Harold Buck <no_one...@attbi.com> wrote:
>
> > > No, he's saying one such sequence ends thus. Actually, there are an
> > > infinite number of sequences that do, but these are not all of the
> > > infinite number of infinite sequences.
> >
> >
> > No, that's not what he's saying. I read it to say "[If you have] An
> > infinitely long string of random digits [it] will have an infinitely
> > long string of 2s in it."
>
> Well, that appears to be the problem. You're wrong. He certainly didn't
> say "*Every* infinitely long string of random digits will have an
> infinitely long string of 2s in it."
The way it's phrased:
"*AN* infinitely long string of random digits *WILL* have...."
implies to me that any time you have one, it will have the property. I
guess I can see that you could interpret it as "There exists one that
has this property," but it would have been clearer if he'd said that.
And, hey, if he meant that, what's the word "random" doing there? If he
just means that there's a string of digits with the property, what does
randomness have to do with it?
Jon and Mary Miller
> If you still disagree, perhaps you can explain what you think a sequence
> looks like that has an infinite string of 2's and also an infinite
> string of 3's.
I can but I won't. Just suffice it to say that I believe in nonstandard
integers, and you can't argue with me about it because of the BoR.
In the BoRing Standard Model that you choose to limit yourself to, you
are, of course, correct.
Jon Miller
Len Berlind
Ree Ludorph <lrud...@panix.com> wrote:
>...with so many undefined terms
>floating around, it's very hard to tell who's saying what, much
>less to say whether the various statements are (a) meaningful,
>(b) true, (c) in contradiction to each other, or (d) all and none
>of the above.
Welcome to AFU.
Harold Buck
It would have avoided a lot of confusion if you would have simply
pointed out that you were talking about something completely different,
which had no bearing on the start of the whole discussion (monkeys
typing and all).
Staffan S
"Nathan Tenny" <n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m> skrev i meddelandet
news:b9v1md$i...@qualcomm.com...
> There are (of course) many infinite sequences of digits that don't contain
> an infinitely long string of 2s. What I can't decide is whether it's
> possible or not for a random process to produce one of them.
It seems to me that if any process that generates random numbers would be a
physical process and because of that subject to the limitations of the
physical world. I.e. for one thing it would not be possible to let it
operate for an infinite period of time, so it wouldn't be possible for it to
produce an infinite sequence of numbers.
Stafafn S
Harold Buck
"Staffan S" <qsw...@passagen.se> wrote:
Random processes exist theoretically, and we can talk about what
"happens" if they run forever without the limitations of practicality.
Kai Henningsen
> R H Draney wrote:
>
> > Just had an interesting thought...suppose you took the monkeys' output and
> > fed it through an automatical spelling corrector before letting anyone see
> > it...would *that* increase the chances that they'd come up with something
> > useful?...r
>
> If you put these monkeys' output through a spelling corrector, wouldn't it
> still be shit?
If you put the original Shakespeare through a spelling corrector, wouldn't
it be shit?
Kai
--
http://www.westfalen.de/private/khms/
"... by God I *KNOW* what this network is for, and you can't have it."
- Russ Allbery (r...@stanford.edu)
Kai Henningsen
> Nathan Tenny <n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m> wrote:
>
> > In article <3ec2e4ae...@news.west.cox.net>,
> > Hatunen <hatu...@cox.net> wrote:
> > >An infinitely long string of random digits will have an
> > >infinitely long string of 2s in it.
> >
> > I can't decide if this is right or not. It certainly is if you add "with
> > probability 1" to the end, but that isn't quite the same as "absolutely".
> >
> > There are (of course) many infinite sequences of digits that don't contain
> > an infinitely long string of 2s. What I can't decide is whether it's
> > possible or not for a random process to produce one of them. It's very
> > *unlikely*, to be sure, but then any *particular* sequence is equally
> > unlikely...and for that reasons, isn't an endless string of 1s (or any
> > other sequence chosen not to have an infinite string of 2s) no *more*
> > unlikely than any other output?
>
> Um, isn't a random process that includes 2s among the digits it
> generates guaranteed to generate an infinite string of 2s if it runs for
> an infinite amount of time?
What if it includes both 2s and 7s? There is no possible string of digits
that contains both an infinite string of 2s and an infinite string of 7s.
(Unless with "an infinite string of 2s" you just mean "infinitely many 2s"
and not "a sequence of infinite length containing only 2s".)
> I would think all possible outputs would
> have to be generated given an infinite amount of time;
Define "an output".
> or to put it
> another way, if there's a finite probability of an event's occuring,
> then given an infinite amount of time, the probability of its *not*
> occuring should be zero.
Sure. The probability of any string of digits of *finite* length is finite
but nonzero; the probability of any such string of infinite length is
zero.
Kai Henningsen
> Yep. An infinite number of them. But that doesn't preclude also
> having an infinite run of 2s. And 3s.
Not at the same time.
Let's describe the sequence as s0 s1 s2 s3 ...
infinite run of digit X ::= exists nX such that for k>nX, sk = X
if n2 >= n3, then sn2 = 2 and sn2 = 3 which can't be. Similarly for
n2 < n3.
Qed.
Kai Henningsen
> An infinitely long string consisting of an infinite number of substrings,
> many being substrings of infinite length containing the digit 2 alone, and
> many containing the digit 3 alone.
Utter nonsense. This is quite impossible.
> Your misunderstanding is that you do not understand the concept of infinity.
As you havejust confused countably infinite with uncountably infinite, you
have just demonstrated that *you* do not understand it *at all*.
Kai Henningsen
> Nope. Back to school for you. You can have that infinitely long string
> of twos, followed by an infinitely long string of threes, followed by an
> infinitely long string of fours, ...
It is trivially obvious that you cannot. Back to school for *you*.
Jon and Mary Miller
Sure it would. You just generate the first number in 1 second, the
second number in 1/2 second, the third in 1/4 second, and so forth. You
have infinitely many numbers in 2 seconds.
Jon Miller
Jon and Mary Miller
> In article <3ec2e4ae...@news.west.cox.net>,
> Hatunen <hatu...@cox.net> wrote:
>
>>An infinitely long string of random digits will have an
>>infinitely long string of 2s in it.
>>
>
> I can't decide if this is right or not. It certainly is if you add "with
> probability 1" to the end, but that isn't quite the same as "absolutely".
>
> There are (of course) many infinite sequences of digits that don't
contain
> an infinitely long string of 2s. What I can't decide is whether it's
> possible or not for a random process to produce one of them.
Sure. Just pick a number "at random". Look at the decimal (or ternary,
what the heck) expansion. If it ends in all 2s, you've found a random
process that produces an infinite string of 2s.
The probability is 0, but that doesn't mean it's impossible.
> It's very
> *unlikely*, to be sure, but then any *particular* sequence is equally
> unlikely...and for that reasons, isn't an endless string of 1s (or any
> other sequence chosen not to have an infinite string of 2s) no *more*
> unlikely than any other output?
Therefore, a truly random process can never generate any output. That's
logic as I know and use it.
Jon "fun with infinity" Miller
John Francis
Kai Henningsen <kaih=8ltKe...@khms.westfalen.de> wrote:
>jo...@panix.com (John Francis) wrote on 14.05.03 in <b9v0ds$hlh$1...@panix5.panix.com>:
>
>> Nope. Back to school for you. You can have that infinitely long string
>> of twos, followed by an infinitely long string of threes, followed by an
>> infinitely long string of fours, ...
>
>It is trivially obvious that you cannot. Back to school for *you*.
Beware of the "trivially obvious" - it's almost always wrong,
and at best means "I can't prove this, so I'll just assert it"
Nathan Tenny
Lee Rudolph <lrud...@panix.com> wrote:
[where "string"=="finite or countable sequence", more or less]
>[...] it is *not* possible to concatenate
>two infinite strings and get a string as the result. You get
>an ordered set of order type \omega + \omega, which is not equal
>to \omega (as an ordinal). On that understanding, you cannot
>have "an infinitely long string of twos, followed by an
>infinitely long string of threes" and fairly say that what you
>have is itself a "string".
Thank you. That's what I wanted to say, only better, so I won't.
I do, though, want to pose a specific question to those who would propose
an infinite string of 2s followed by an infinite string of 3s: What's the
index of the first 3?
>Oh, and I never got a master's degree.
How'd that happen? No safety-net master's around qualifying-exam time?
We did that partly to avoid throwing people out emptyhanded, I think,
on grounds that getting to the qual stage merited more qualifications
than never going to gradual school a-tall did.
NT
--
Nathan Tenny | When the world ends, there'll be no more
Qualcomm, Inc., San Diego, CA | air. That's why it's important to pollute
<nten...@qualcomm.com> | the air now. Before it's too late.
| -- Kathy Acker
John Francis
Nathan Tenny <nten...@qualcomm.com> wrote:
>In article <ba0ltf$221$1...@panix2.panix.com>,
>Lee Rudolph <lrud...@panix.com> wrote:
>[where "string"=="finite or countable sequence", more or less]
>>[...] it is *not* possible to concatenate
>>two infinite strings and get a string as the result. You get
>>an ordered set of order type \omega + \omega, which is not equal
>>to \omega (as an ordinal). On that understanding, you cannot
>>have "an infinitely long string of twos, followed by an
>>infinitely long string of threes" and fairly say that what you
>>have is itself a "string".
>
>Thank you. That's what I wanted to say, only better, so I won't.
>
>I do, though, want to pose a specific question to those who would propose
>an infinite string of 2s followed by an infinite string of 3s: What's the
>index of the first 3?
How about omega/2 ?
Although, as pointed out, we're getting away from "strings", here.
Hatunen
wrote:
>In article <8ltKe...@khms.westfalen.de>,
>Kai Henningsen <kaih=8ltKe...@khms.westfalen.de> wrote:
>>jo...@panix.com (John Francis) wrote on 14.05.03 in <b9v0ds$hlh$1...@panix5.panix.com>:
>>
>>> Nope. Back to school for you. You can have that infinitely long string
>>> of twos, followed by an infinitely long string of threes, followed by an
>>> infinitely long string of fours, ...
>>
>>It is trivially obvious that you cannot. Back to school for *you*.
>
>Beware of the "trivially obvious" - it's almost always wrong,
>and at best means "I can't prove this, so I'll just assert it"
It is a fundamental of the lowest infinity that if you add two
infinite numbers you get an infinite number.
************* DAVE HATUNEN (hat...@cox.net) *************
* Tucson Arizona, out where the cacti grow *
* My typos & mispellings are intentional copyright traps *
Hatunen
(Nathan Tenny) wrote:
>In article <ba0ltf$221$1...@panix2.panix.com>,
>Lee Rudolph <lrud...@panix.com> wrote:
>[where "string"=="finite or countable sequence", more or less]
>>[...] it is *not* possible to concatenate
>>two infinite strings and get a string as the result. You get
>>an ordered set of order type \omega + \omega, which is not equal
>>to \omega (as an ordinal). On that understanding, you cannot
>>have "an infinitely long string of twos, followed by an
>>infinitely long string of threes" and fairly say that what you
>>have is itself a "string".
>
>Thank you. That's what I wanted to say, only better, so I won't.
>
>I do, though, want to pose a specific question to those who would propose
>an infinite string of 2s followed by an infinite string of 3s: What's the
>index of the first 3?
When dealing with infinities that's not a proper question since
it is equivalent to asking what the first number beyond infinity
is, i.e., what is infinity + 1?
Nathan Tenny
Binyamin Dissen <post...@dissensoftware.com> wrote:
>On Thu, 15 May 2003 14:52:27 GMT Harold Buck <no_one...@attbi.com> wrote:
>>If you still disagree, perhaps you can explain what you think a sequence
>>looks like that has an infinite string of 2's and also an infinite
>>string of 3's.
>
>
>An infinitely long string consisting of an infinite number of substrings, many
>being substrings of infinite length containing the digit 2 alone, and many
>containing the digit 3 alone.
I'll pose you the same question I just floated on another branch of this
thread: What's the index of the first 3?
If you have an answer for that, what exactly do you mean by the words
"sequence" and "string"?
>Your misunderstanding is that you do not understand the concept of infinity.
I distrust anyone who describes it as one concept.
N"Cite: Me."T
Harold Buck
hatu...@cox.net (Hatunen) wrote:
> On 15 May 2003 20:23:58 -0400, jo...@panix.com (John Francis)
> wrote:
>
> >In article <8ltKe...@khms.westfalen.de>,
> >Kai Henningsen <kaih=8ltKe...@khms.westfalen.de> wrote:
> >>jo...@panix.com (John Francis) wrote on 14.05.03 in
> >><b9v0ds$hlh$1...@panix5.panix.com>:
> >>
> >>> Nope. Back to school for you. You can have that infinitely long string
> >>> of twos, followed by an infinitely long string of threes, followed by an
> >>> infinitely long string of fours, ...
> >>
> >>It is trivially obvious that you cannot. Back to school for *you*.
> >
> >Beware of the "trivially obvious" - it's almost always wrong,
> >and at best means "I can't prove this, so I'll just assert it"
>
> It is a fundamental of the lowest infinity that if you add two
> infinite numbers you get an infinite number.
Oh, I'm sure everyone agrees with this. The problem is you can't make a
list (=string) of numbers that has an infinite run of 2's followed by an
infinite run of 3's.
But, sure, if you just dump them all in a box w/o regard to order, you
get infinity + infinity = infinity digits.
Lars Eighner
<8ltKd...@khms.westfalen.de>,
the lovely and talented Kai Henningsen
broadcast on alt.folklore.urban:
> What if it includes both 2s and 7s? There is no possible string of digits
> that contains both an infinite string of 2s and an infinite string of 7s.
> (Unless with "an infinite string of 2s" you just mean "infinitely many 2s"
> and not "a sequence of infinite length containing only 2s".)
Sorry, an infinite string of random digits will contain sequences of
infinite length containing only 2s and another one containing only 7s.
You haven't really studied transfinite numbers, have you? Here's
a little exercise to get you going: you are the desk clerk at the
infinite hotel which is completely booked up. An infinite number of
new guests arrive. How do find rooms for all of them? (There are
many ways.)
--
Lars Eighner -finger for geek code- eig...@io.com http://www.io.com/~eighner/
I have trouble making the sequence "if you are in the Mongolian desert,
you might be in luck" make any sort of semantic sense.... --R H Draney
TeaLady (Mari C.)
news:no_one_knows-49E2...@netnews.attbi.com:
Hell, to me *all* numbers are just a subset of an infinite
number of strings of infinite subsets of infinite sets of infinite
numbers.
It's just one big number all broken up into littler ones.
I just wish it would break off some more zeros and tack them onto
my bank balance, to the left of the decimal and right of the 1st
non-zero positive integer. A group of about 5, or even 6, would be
nice.
--
Tea"Math is one big muddle to me"Lady (mari)
...But now I'm feeling so much better, I could cakewalk into town.
David Wnsemius
> In article <ba0ltf$221$1...@panix2.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
> [where "string"=="finite or countable sequence", more or less]
>>[...] it is *not* possible to concatenate
>>two infinite strings and get a string as the result. You get
>>an ordered set of order type \omega + \omega, which is not equal
>>to \omega (as an ordinal). On that understanding, you cannot
>>have "an infinitely long string of twos, followed by an
>>infinitely long string of threes" and fairly say that what you
>>have is itself a "string".
>
> Thank you. That's what I wanted to say, only better, so I won't.
>
> I do, though, want to pose a specific question to those who would propose
> an infinite string of 2s followed by an infinite string of 3s: What's the
> index of the first 3?
>
If we leave aside the random aspect of constructing strings and create a
string with sequential additions (appendations?) of sequences of "2" and
"7" of increasing lengths, will we not have a string that has subsrings
that meet the desired specifications? The nth concatenated substring will
be n in length. A 2's substring will be incorporated eventually to exceed
any specified length, meeting the requirement to be countably infinite.
I am not so sure that the random string can be guaranteed to contain such a
substring because there is no deterministic aspect to the method of
construction. It would seem that as the length of the string approached
(countable) infinity the probability that a random string of length n would
be all 2's would approach zero.
--
David "1st year college math for me" Winsemius
David Wnsemius
> Sure it would. You just generate the first number in 1 second, the
> second number in 1/2 second, the third in 1/4 second, and so forth.
> You have infinitely many numbers in 2 seconds.
>
One is generally supposed to insert keywords when trolling.
--
David "" Winsemius
Harold Buck
David Wnsemius <dwin$emiu$@attbi.com.not> wrote:
> If we leave aside the random aspect of constructing strings and create a
> string with sequential additions (appendations?) of sequences of "2" and
> "7" of increasing lengths, will we not have a string that has subsrings
> that meet the desired specifications? The nth concatenated substring will
Sure, infinite *sub*strings are no problem, but that wasn't what the guy
was talking about. And if you're okay with substrings, give me any long
random string of ASCII characters and I'll start pulling Shakespeare out
of it for you.
Aaron Davies
> In our last episode,
> <8ltKd...@khms.westfalen.de>,
> the lovely and talented Kai Henningsen
> broadcast on alt.folklore.urban:
>
> > What if it includes both 2s and 7s? There is no possible string of digits
> > that contains both an infinite string of 2s and an infinite string of 7s.
> > (Unless with "an infinite string of 2s" you just mean "infinitely many 2s"
> > and not "a sequence of infinite length containing only 2s".)
>
> Sorry, an infinite string of random digits will contain sequences of
> infinite length containing only 2s and another one containing only 7s.
>
> You haven't really studied transfinite numbers, have you? Here's
> a little exercise to get you going: you are the desk clerk at the
> infinite hotel which is completely booked up. An infinite number of
> new guests arrive. How do find rooms for all of them? (There are
> many ways.)
That reminds me, are you familiar with the filk song "Banned from
Aleph"? <http://www-users.cs.york.ac.uk/~susan/sf/filk/aleph.htm> Quick
precis: what if Aleph1 people show up at a hotel with only Aleph0 rooms?
--
__ __
/ ) / )
/--/ __. __ ________ / / __. , __o _ _
/ (_(_/|_/ (_(_) / / <_ /__/_(_/|_\/ <__</_/_)_
Barbara Needham
>
> You haven't really studied transfinite numbers, have you? Here's
> a little exercise to get you going: you are the desk clerk at the
> infinite hotel which is completely booked up. An infinite number of
> new guests arrive. How do find rooms for all of them? (There are
> many ways.)
The infinite hotel burned down. I just read it. So it must be true.
http://www.c3.lanl.gov/mega-math/workbk/infinity/inhotel.html
For others who would like to be enlightened [or not.]
--
Barbara Needham
Hatunen
(Nathan Tenny) wrote:
>In article <chb7cvofg7jib1rvk...@4ax.com>,
>Binyamin Dissen <post...@dissensoftware.com> wrote:
>>On Thu, 15 May 2003 14:52:27 GMT Harold Buck <no_one...@attbi.com> wrote:
>>>If you still disagree, perhaps you can explain what you think a sequence
>>>looks like that has an infinite string of 2's and also an infinite
>>>string of 3's.
>>
>>Simple.
>>
>>An infinitely long string consisting of an infinite number of substrings, many
>>being substrings of infinite length containing the digit 2 alone, and many
>>containing the digit 3 alone.
>
>I'll pose you the same question I just floated on another branch of this
>thread: What's the index of the first 3?
And I'll give you the same answer: it's a nonsense question when
dealing with the transfinites.
David Wnsemius
news:no_one_knows-D343...@netnews.attbi.com:
> In article <Xns937CE8BF2101Ed...@63.240.76.16>,
> David Wnsemius <dwin$emiu$@attbi.com.not> wrote:
>
>> If we leave aside the random aspect of constructing strings and
>> create a string with sequential additions (appendations?) of
>> sequences of "2" and "7" of increasing lengths, will we not have a
>> string that has subsrings that meet the desired specifications? The
>> nth concatenated substring will
>
>
> Sure, infinite *sub*strings are no problem, but that wasn't what the
> guy was talking about. And if you're okay with substrings, give me any
> long random string of ASCII characters and I'll start pulling
> Shakespeare out of it for you.
>
It appeared to me that many contributors, yourself included, WERE having a
problem with infinite subsrings. My effort was to specify a construction
and explicit ordering. I placed my effort following Nathan Tenny's
question:
> I do, though, want to pose a specific question to those who
> would propose an infinite string of 2s followed by an infinite
> string of 3s: What's the index of the first 3?
I have not precisely answered Nathan's question because it cannot be
answered, but I have allowed an answer to the question: What is the index
of the first 3 that follows the string of any arbitrarily large length of
2's. And but it also addreses yours in:
<no_one_knows-1BB4...@netnews.attbi.com>
> If you still disagree, perhaps you can explain what you think
> a sequence looks like that has an infinite string of 2's and
> also an infinite string of 3's.
--
David "Buzz Lightyear" Winsemius
buttoned down mind
Schmitt) wrote:
>Today's Times contains a report about an attempt to demonstrate
>the monkeys-typewriters-Shakespeare theory by experiment. If you
>go to their homepage and search using the obvious terms in the
>near future, you can read an article which will, at the very
>least, make the corners of you mouth turn upwards. It is also at
>the BBC site:
>
>http://news.bbc.co.uk/1/hi/england/devon/3013959.stm
>
>Admittedly it is a very small sample, with hardly enough monkeys,
>typewriters or time, but it is heartening to hear that bashing a
>horse until only a faint stain remains is still a sport academia
>supports.
>
>John "a monkey barred" Schmitt
>
>
> --
>If you have nothing to say, or rather, something extremely stupid
>and obvious, say it, but in a 'plonking' tone of voice - i.e.
>roundly, but hollowly and dogmatically. - Stephen Potter
To be or not to be, that is the gazornanplat...
Harold Buck
David Wnsemius <dwin$emiu$@attbi.com.not> wrote:
> Harold Buck wrote in
> news:no_one_knows-D343...@netnews.attbi.com:
>
> > In article <Xns937CE8BF2101Ed...@63.240.76.16>,
> > David Wnsemius <dwin$emiu$@attbi.com.not> wrote:
> >
> >> If we leave aside the random aspect of constructing strings and
> >> create a string with sequential additions (appendations?) of
> >> sequences of "2" and "7" of increasing lengths, will we not have a
> >> string that has subsrings that meet the desired specifications? The
> >> nth concatenated substring will
> >
> >
> > Sure, infinite *sub*strings are no problem, but that wasn't what the
> > guy was talking about. And if you're okay with substrings, give me any
> > long random string of ASCII characters and I'll start pulling
> > Shakespeare out of it for you.
> >
> It appeared to me that many contributors, yourself included, WERE having a
> problem with infinite subsrings. My effort was to specify a construction
> and explicit ordering. I placed my effort following Nathan Tenny's
> question:
Definitional problem, again. When I say "substring" I mean "not
necessarily consecutive," so that 121212121212.... has an infinite
substring of 2's, and of 1's. But the standard model does not allow you
to have 1........12......, where you have an infinite number of 1's
followed by an infinite number of 2's *in sequence*.
I don't doubt that there's some branch of math to deal with this, but it
has nothing to do with the discussion that started this thread, namely a
monkey pounding on a keyboard.
Charles A Lieberman
kaih=8ltKd...@khms.westfalen.de (Kai Henningsen) wrote:
> > If you put these monkeys' output through a spelling corrector, wouldn't it
> > still be shit?
>
> If you put the original Shakespeare through a spelling corrector, wouldn't
> it be shit?
Sure, an *English* spellcheck.
Charles "merging with the Klingon threads" Lieberman
--
Charles A. Lieberman | "Granted, the animals without heads, bones, or
Brooklyn, NY, USA | limbs need a lot of assistance to breed, but so
cali...@bigfoot.com | what?" Nathan Tenny teaches AFU animal husbandry
Dr H
On 13 May 2003, R H Draney wrote:
}In article <Pine.GSU.4.21.03051...@garcia.efn.org>, Dr says...
}>
}> Random sequences can, and frequently do include long sections of a
}> single value. The sequences are no less random for the fact that
}> we tend to notice groups of similar items and /assume/ they represent
}> a deliberate pattern.
}
}I see it more as evidence that monkeys are, like human beings, incapable of
}behaving in a truly random fashion...I thought the quote was a little pithier
}(and said something about human beings being unable to create a random
}sequence), but from Knuth's "The Art of Computer Programming: Volume 2,
}Seminumerical Algorithms (3rd edition)", it takes the form of the first exercise
}in the section on random-number generators:
}
}1. [20] Suppose that you wish to obtain a decimal digit at random, not using a
}computer. Which of the following methods would be suitable?
}...
}f) Ask a friend to think of a random digit, and use the digit he names.
}g) Ask an enemy to think of a random digit, and use the digit he names.
}...
}
}From the Answers section:
}...(f,g) No. People usually think of certain digits (like 7) with higher
}probability....
}
}To which I might add: "monkeys usually hit certain keys on a typewriter (like S)
}with higher probability"...we now have empirical evidence for the latter....r
While discussing this experiment with my brother he pointed out that
Shakespeare did, in fact, employ the letter "s" many, many times in
his works. Hence the experiment was a success; the reserachers just
misinterpreted their data.
Dr H
Dr H
On Thu, 15 May 2003, Harold Buck wrote:
}In article <Pine.GSU.4.21.03051...@garcia.efn.org>,
} Dr H <hiaw...@efn.org> wrote:
}
}>
}> Random sequences can, and frequently do include long sections of a
}> single value. The sequences are no less random for the fact that
}> we tend to notice groups of similar items and /assume/ they represent
}> a deliberate pattern.
}
}And such coincidences are the basis of much of the "paranormal": people
}are very good at finding patterns in random data, and then they feel the
}need to "explain" it.
Sure. But humans in general seem to be wired to recognize certain
things as "patterns". Most people who look at a string of randomly
generated nubmers that contains the substring "1234" are going to
latch onto that as a "meaningful" pattern. It's not the perception
of pattern that's the problem, but the ascribing of "meaning" to
the pattern.
Dr H
Karen J. Cravens
news:no_one_knows-40EE...@netnews.attbi.com:
> Definitional problem, again. When I say "substring" I mean "not
> necessarily consecutive," so that 121212121212.... has an infinite
> substring of 2's, and of 1's. But the standard model does not allow
> you to have 1........12......, where you have an infinite number of
> 1's followed by an infinite number of 2's *in sequence*.
It's been a lot of years since DiffEq and all those courses involving
brane-hurting infinity-related math, so I'll have to ask: why not?
--
Karen J. Cravens
Dr H
On 14 May 2003, Nathan Tenny wrote:
}Isn't that the opposite of what you said at the beginning? If "every
}sequence you can think of is in there" (and I agree with that), then
}all the infinitely many sequences *without* an infinite run of 2s are
}"in there", where "in there" means "possible output of a random process".
}
}Now I've confused myself even more by mentioning measures.
Welcome to the wonderful world of transfinites. :-)
Dr H
Dr H
On Thu, 15 May 2003, Harold Buck wrote:
}In article <3ec2e4ae...@news.west.cox.net>,
} hatu...@cox.net (Hatunen) wrote:
}
}>
}> An infinitely long string of random digits will have an
}> infinitely long string of 2s in it.
}
}
}It's unclear if you're joking, but I hope so. There is no upper bound to
}the length of a run of 2's in the sequence, but the only way you can
}have an infinitely long sequence of 2's is if it ends with a string of
}repeating 2's.
Actually, no. An /infinitely/ long string of number can contain any
number of infinitely long subsets.
Dr H
Dr H
On Thu, 15 May 2003, Hatunen wrote:
}On 14 May 2003 20:31:25 -0700, n_t_e_nn_y_@q_ual_c_o_m_m_.c_o_m
}(Nathan Tenny) wrote:
}
}>What I can't decide is whether it's
}>possible or not for a random process to produce one of them. It's very
}>*unlikely*, to be sure, but then any *particular* sequence is equally
}>unlikely...
}
}And equally likely. In fact, every sequence you can think of will
}be in there.
And if you put an infinite number on monkeys in a room with an
infinite number of 10-key machines...
Dr H
Dr H
On 15 May 2003, Nathan Tenny wrote:
}In article <chb7cvofg7jib1rvk...@4ax.com>,
}Binyamin Dissen <post...@dissensoftware.com> wrote:
}>On Thu, 15 May 2003 14:52:27 GMT Harold Buck <no_one...@attbi.com> wrote:
}>>If you still disagree, perhaps you can explain what you think a sequence
}>>looks like that has an infinite string of 2's and also an infinite
}>>string of 3's.
}>
}>Simple.
}>
}>An infinitely long string consisting of an infinite number of substrings, many
}>being substrings of infinite length containing the digit 2 alone, and many
}>containing the digit 3 alone.
}
}I'll pose you the same question I just floated on another branch of this
}thread: What's the index of the first 3?
The index of the first 3 is obviously "infinity + 1".
Dr H
David Wnsemius
> Definitional problem, again. When I say "substring" I mean "not
> necessarily consecutive," so that 121212121212.... has an infinite
> substring of 2's, and of 1's. But the standard model does not allow you
> to have 1........12......, where you have an infinite number of 1's
> followed by an infinite number of 2's *in sequence*.
>
I explained how that is incorrect. I described how you could construct an
infinite sequence that would include a countably infinite subsequence of
adjacent 2's . It is a similar exercise to showing that the set of rational
numbers is of the same ordinality as the integers.
> I don't doubt that there's some branch of math to deal with this, but it
> has nothing to do with the discussion that started this thread, namely a
> monkey pounding on a keyboard.
>
And you snipped the second of my paragraphs where I discussed the problem
of random sequences, and how I thought the problems were different.
--
David "will construct aleph null sets for food" Winsemius