Cryptography-Digest Digest #718, Volume #13 Mon, 19 Feb 01 21:13:01 EST
Contents:
Re: Given any arbitrary numbers a and b. (Dave Seaman)
block ciphers (neXussT)
Re: block ciphers ("Paul Pires")
Re: What's a KLB-7? (Steven James Forsberg)
Re: What's a KLB-7? (David Hamer)
Re: A seriously different cipher concept (long) (Mok-Kong Shen)
2.6.3ckt ("Ryan M. McConahy")
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a (jtnews)
Re: block ciphers (neXussT)
Re: Given any arbitrary numbers a and b. ([EMAIL PROTECTED])
Re: Given any arbitrary numbers a and b.Can I ALWAYS find a (Mok-Kong Shen)
Re: A seriously different cipher concept (long) ("Paul Pires")
Re: Rnadom Numbers (Jerry Coffin)
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental number
between a and b? (Jerry Coffin)
Re: block ciphers ("Paul Pires")
Re: block ciphers ("Paul Pires")
Re: block ciphers ("Paul Pires")
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental number
between a and b? (Randy Poe)
Re: block ciphers (neXussT)
Re: (Just the facts) ("Paul Pires")
Re: Is there an algorithm to sequentially enumerate all transcendental (Bryan
Olson)
Re: Is there an algorithm to sequentially enumerate all transcendental (Bryan
Olson)
Re: Given any arbitrary numbers a and b.Can I ALWAYS find a transcendental
numberbetween a and b? (John Savard)
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental number
between a and b? (John Savard)
Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental number
between a and b? (John Savard)
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
(John Savard)
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
(John Savard)
Re: Is there an algorithm to sequentially enumerate all transcendental numbers?
("Dik T. Winter")
----------------------------------------------------------------------------
From: [EMAIL PROTECTED] (Dave Seaman)
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b.
Date: 19 Feb 2001 15:43:25 -0500
In article <[EMAIL PROTECTED]>,
jtnews <[EMAIL PROTECTED]> wrote:
>Since my previous post seemed to be not to clear
>to some people. Let me restate the problem another
>way which I hope will be simpler to understand.
>Given any arbitrary numbers a and b where a and
>b are NOT the same number.
>Can I ALWAYS find a transcendental number
>between a and b?
I don't see how this is going to have any effect on your previous
question, but the answer is yes. Given any two distinct real numbers a
and b (transcendental or not), there is a transcendental number lying
between them. We say that the transcendentals are dense in the real
line.
Proof: We may as well assume that a < b. Thus the interval (a,b) has
positive measure. The set of algebraic numbers in (a,b), being
countable, has measure 0. Therefore, the transcendentals in (a,b) have
positive measure.
By the way, the rationals are also dense in the real line, despite being
countable. Density has nothing to do with countability.
--
Dave Seaman [EMAIL PROTECTED]
Amnesty International calls for new trial for Mumia Abu-Jamal
<http://www.amnestyusa.org/abolish/reports/mumia/>
------------------------------
From: [EMAIL PROTECTED] (neXussT)
Subject: block ciphers
Date: Mon, 19 Feb 2001 21:09:41 GMT
i'm using borland c++ and would like to know how to create a block
cipher in c++
i've tried a few different things, but nothing works. does nebody
know of a link, or have information themselves on how to do this?
thx
neXussT
------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 13:23:06 -0800
neXussT <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED]...
> i'm using borland c++ and would like to know how to create a block
> cipher in c++
>
> i've tried a few different things, but nothing works. does nebody
> know of a link, or have information themselves on how to do this?
> thx
> neXussT
Not as easy as it looks?
You are asking for the equivalent of "how to build a nuclear reactor"
It's not a subject that can be addressed in a few posts and many
replies you are likely to get are going to be in the nature of
"Don't try it"
I would suggest that you read this news groups FAQ and
read some introductory material on the subject.
A suggestion is Applied Cryptography 2cnd edition by
Bruce Schneier.
Paul
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------------------------------
From: [EMAIL PROTECTED] (Steven James Forsberg)
Subject: Re: What's a KLB-7?
Date: 19 Feb 2001 21:35:32 GMT
: 2001-02-19 17:29:00 GMT
: On board the H.M.S. Belfast today (a heavy-light cruiser
: museum piece moored in the London Pool) in the Electronic
: Warfare Shack, behind glass, I observed an obvious piece
: of crypto equipment which proclaimed itself an instance
: of NSA "KLB-7/T SEC" S#12405, best-delivery-before date
: sometime in 1990 to somewhere in Whitehall.
: Can anyone say what this hardware was for? The Belfast
: saw 'angry' service up to the end of the Korean War.
This is probably a version of the KL-7 "Adonis" offline
encryption system. Take a look at the following page to be certain:
http://webhome.idirect.com/~jproc/crypto/kl7.htm
regards,
====================================
[EMAIL PROTECTED]
------------------------------
Date: Mon, 19 Feb 2001 16:45:03 -0500
From: David Hamer <[EMAIL PROTECTED]>
Subject: Re: What's a KLB-7?
Richard Outerbridge wrote:
>
> -----BEGIN PGP SIGNED MESSAGE-----
>
> 2001-02-19 17:29:00 GMT
> On board the H.M.S. Belfast today (a heavy-light cruiser
> museum piece moored in the London Pool) in the Electronic
> Warfare Shack, behind glass, I observed an obvious piece
> of crypto equipment which proclaimed itself an instance
> of NSA "KLB-7/T SEC" S#12405, best-delivery-before date
> sometime in 1990 to somewhere in Whitehall.
>
> Can anyone say what this hardware was for? The Belfast
> saw 'angry' service up to the end of the Korean War.
>
> outer
Richard...
It's a KL-7. There is a fair amount of information on
this machine [with photographs] on Jerry Proc's page:
<http://webhome.idirect.com/~jproc/crypto/kl7.html>
KL-7 is also included, briefly, in 'Machine Cryptography
and Modern Cryptanalysis by C.A. Deavours & Louis Kruh,
Artech House, 1985'
DHH
--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
David Hamer The Crypto Simulation Group
[EMAIL PROTECTED] or [EMAIL PROTECTED]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
------------------------------
From: Mok-Kong Shen <[EMAIL PROTECTED]>
Subject: Re: A seriously different cipher concept (long)
Date: Mon, 19 Feb 2001 22:46:31 +0100
Paul Pires wrote:
>
> The following is a description of a cipher concept I
> have been working on for quite awhile. It's reason
> for existence is to explore a different method that
> has the potential for being unusually fast. All rights
> are preserved. Some or all of this material may be
> covered by patents held by the author or unknown
> others. No presentment is made that the methodology
> contained herein is suitable, safe or unencumbered for
> commercial use.
[snip]
I am aware that I could be asking for much too much, but
I have the impression that it would be fine for better
understanding of most readers if you could give in
addition a contigeous piece of pseudo-code that is
extremely compact yet not omitting the essentials of
the algorithm (with some terse comments/pointers relating
to the texts in the article).
M. K. Shen
------------------------------
From: "Ryan M. McConahy" <[EMAIL PROTECTED]>
Subject: 2.6.3ckt
Date: Mon, 19 Feb 2001 16:49:36 -0500
Why is it that PGP 2.6.3ckt dosn't work under DOS???
Ryan M. McConahy
------------------------------
Date: Mon, 19 Feb 2001 16:54:34 -0500
From: jtnews <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a
Paul Lutus wrote:
>
> "jtnews" <[EMAIL PROTECTED]> wrote in message
> news:[EMAIL PROTECTED]...
> > Since my previous post seemed to be not to clear
> > to some people. Let me restate the problem another
> > way which I hope will be simpler to understand.
> >
> > Given any arbitrary numbers a and b where a and
> > b are NOT the same number.
> >
> > Can I ALWAYS find a transcendental number
> > between a and b?
>
> If A and B are integers, yes.
>
> 1. Pi is transcendental.
> 2. Therefore, by definition, Pi+n is transcendental if n = integer.
> 3. Therefore this sequence can be constructed:
>
> A, A+Pi-3, B.
>
> As usual, I am more than happy to be corrected on this sequence of steps by
> those more skilled than I am.
>
> Is this the answer to your question?
Not exactly, any proof must work for any arbitrary
value of a and b.
Jan Kristian Haugland 's answer of
>Yes. If c is a rational number with 0 < c < b-a,
>then one of (a+b)/2, (a+b)/2 + c/pi is transcendental.
>This, however, doesn't show that the set of
>transcendentals is uncountable (see other comments
>about the rationals).
is closest to what I was looking for.
However, I'm not sure what uncountable
and countable mean. I'm going to have
to lookup these terms myself.
------------------------------
From: [EMAIL PROTECTED] (neXussT)
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 21:59:13 GMT
ok..sounds resoanable...
i was just looking for a way to make it so that my enkryption could
not be dekrypted by looking for occurences of characters
such as:
slkfdj.;lkj.oiwejf.dflsj..lskjoei.
^ ^ ^ ^^ ^
neXussT
>
>Not as easy as it looks?
>
>You are asking for the equivalent of "how to build a nuclear reactor"
>It's not a subject that can be addressed in a few posts and many
>replies you are likely to get are going to be in the nature of
>"Don't try it"
>
>I would suggest that you read this news groups FAQ and
>read some introductory material on the subject.
>A suggestion is Applied Cryptography 2cnd edition by
>Bruce Schneier.
>
>Paul
------------------------------
From: [EMAIL PROTECTED]
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b.
Date: 19 Feb 2001 21:45:51 GMT
In article <[EMAIL PROTECTED]>, jtnews
<[EMAIL PROTECTED]> writes:
>Since my previous post seemed to be not to clear
>to some people. Let me restate the problem another
>way which I hope will be simpler to understand.
>Given any arbitrary numbers a and b where a and
>b are NOT the same number.
>Can I ALWAYS find a transcendental number
>between a and b?
Well, such numbers certainly exist. In fact, given
any transcendental number (say pi), there are infinitely
many nonzero rational multiples of that number between a
and b. If, e.g. 0 < a < b, then any rational number r for
which 0 < a/pi < b/pi will satisfy a < r*pi < b.
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------------------------------
From: Mok-Kong Shen <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b.Can I ALWAYS find a
Date: Mon, 19 Feb 2001 23:01:06 +0100
Dave Seaman wrote:
>
[snip]
> Given any two distinct real numbers a
> and b (transcendental or not), there is a transcendental number lying
> between them. We say that the transcendentals are dense in the real
> line.
>
> Proof: We may as well assume that a < b. Thus the interval (a,b) has
> positive measure. The set of algebraic numbers in (a,b), being
> countable, has measure 0. Therefore, the transcendentals in (a,b) have
> positive measure.
Wouldn't it be possible to show that, given any (specific)
transcendental number T, there is always a rational number
R such that a < R*T < b? (I presume that Mr. jtnews
would have liked this much better.)
M. K. Shen
------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: A seriously different cipher concept (long)
Date: Mon, 19 Feb 2001 14:04:29 -0800
Mok-Kong Shen <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
<Snip>
> I am aware that I could be asking for much too much, but
> I have the impression that it would be fine for better
> understanding of most readers if you could give in
> addition a contigeous piece of pseudo-code that is
> extremely compact yet not omitting the essentials of
> the algorithm (with some terse comments/pointers relating
> to the texts in the article).
I thought that is what I did in Section 4, The detailed
walk-through. It's about as "Terse and Pseudo" as I
know how to be. Is there some particular problem
or lack of clarity I might help with? Is it all garbled
or just part?
Thanks,
Paul
------------------------------
From: Jerry Coffin <[EMAIL PROTECTED]>
Subject: Re: Rnadom Numbers
Date: Mon, 19 Feb 2001 15:21:32 -0700
In article <96rp86$2p4$[EMAIL PROTECTED]>, Simon.Johnson6
@btinternet.com says...
> Would I right in saying that if a crypto-secure pseudo-random generator was
> unable to be cryptanalysed until x bytes of stream were recovered then all
> of the bytes previous to the x'th byte must be statistically random?
It depends a bit on what you mean by "cryptanalyzed" in this case.
With a PRNG, you've got a number of possibilities. You might be able
to find _some_ correlation with a relatively small amount of input,
but require considerably more input before you could accurately
predict the next number it was going to produce.
In the end, about all testing can do is say that it found a
correlation or it didn't. If testing didn't find correlation, it
does NOT mean that none can exist -- a different sort of test might
be able to find a correlation that your testing missed.
--
Later,
Jerry.
The Universe is a figment of its own imagination.
------------------------------
From: Jerry Coffin <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b?
Date: Mon, 19 Feb 2001 15:33:10 -0700
In article <[EMAIL PROTECTED]>,
[EMAIL PROTECTED] says...
[ ... ]
> Given any arbitrary numbers a and b where a and
> b are NOT the same number.
>
> Can I ALWAYS find a transcendental number
> between a and b?
Yes.
--
Later,
Jerry.
The Universe is a figment of its own imagination.
------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 14:13:14 -0800
neXussT <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED]...
> ok..sounds resoanable...
> i was just looking for a way to make it so that my enkryption could
> not be dekrypted by looking for occurences of characters
> such as:
> slkfdj.;lkj.oiwejf.dflsj..lskjoei.
> ^ ^ ^ ^^ ^
Well, your instinct was right. A block cipher won't have that
problem but there is a lot of learning ahead. How they work
and how they fail is facinating and well worth the study.
Paul
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------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 14:13:14 -0800
neXussT <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED]...
> ok..sounds resoanable...
> i was just looking for a way to make it so that my enkryption could
> not be dekrypted by looking for occurences of characters
> such as:
> slkfdj.;lkj.oiwejf.dflsj..lskjoei.
> ^ ^ ^ ^^ ^
Well, your instinct was right. A block cipher won't have that
problem but there is a lot of learning ahead. How they work
and how they fail is facinating and well worth the study.
Paul
====== Posted via Newsfeeds.Com, Uncensored Usenet News ======
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======= Over 80,000 Newsgroups = 16 Different Servers! ======
------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 14:13:14 -0800
neXussT <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED]...
> ok..sounds resoanable...
> i was just looking for a way to make it so that my enkryption could
> not be dekrypted by looking for occurences of characters
> such as:
> slkfdj.;lkj.oiwejf.dflsj..lskjoei.
> ^ ^ ^ ^^ ^
Well, your instinct was right. A block cipher won't have that
problem but there is a lot of learning ahead. How they work
and how they fail is facinating and well worth the study.
Paul
====== Posted via Newsfeeds.Com, Uncensored Usenet News ======
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======= Over 80,000 Newsgroups = 16 Different Servers! ======
------------------------------
From: [EMAIL PROTECTED] (Randy Poe)
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b?
Date: Mon, 19 Feb 2001 22:04:05 GMT
On Mon, 19 Feb 2001 15:01:18 -0500, jtnews <[EMAIL PROTECTED]>
wrote:
>Since my previous post seemed to be not to clear
>to some people. Let me restate the problem another
>way which I hope will be simpler to understand.
>
>Given any arbitrary numbers a and b where a and
>b are NOT the same number.
>
>Can I ALWAYS find a transcendental number
>between a and b?
This property of the set of transcendental numbers is called "dense",
and as is pointed out elsewhere, the answer is yes, the
transcendentals are dense on the reals. But dense does not imply
uncountable.
The rationals are also dense: You can always find a rational number
between any two distinct real numbers a, b. In fact you can find
infinitely many rationals between a and b.
- Randy
------------------------------
From: [EMAIL PROTECTED] (neXussT)
Subject: Re: block ciphers
Date: Mon, 19 Feb 2001 22:56:44 GMT
thx paul,
i'll c what more i can find about them
neXussT
------------------------------
From: "Paul Pires" <[EMAIL PROTECTED]>
Subject: Re: (Just the facts)
Date: Mon, 19 Feb 2001 17:11:44 -0800
Mok-Kong Shen <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]...
<snip>
> I am aware that I could be asking for much too much, but
> I have the impression that it would be fine for better
> understanding of most readers if you could give in
> addition a contigeous piece of pseudo-code that is
> extremely compact yet not omitting the essentials of
> the algorithm (with some terse comments/pointers relating
> to the texts in the article).
Ok, Here goes.
Plaintext, chiphertext and State table B are all handled
as a block of 64 units or "elements". State A is a scrambled
table of 0-63 values.
"i" is the element counter and is used mod 64 .
To process a block:
1, Encrypt plaintext:
C[A[(i +32)] = B[i] xor P[A[i]];
2, Change all elements in State B;
B[i] = (B[i]>>5) xor (B[i+1]<<27) xor P[A[63-i]] xor C[i];
3, Select a value in B to serve as a change argument "Tsel" for A.
Tsel = B[A[x mod 64]] mod 64;
4, Change all elements in state A according to Tsel; (Sorry, this
is a really simple action that is real hard to explain simply. All it does
is move all the values in state A to new locations in one of 64
different ways. The "way" is dictated by Tsel from above. "Trans"
is a statis table define different transpositions that can be used)
"DW_A" is a 32-bit pointer,
"BT_A" is an 8-bit pointer,
"&" denotes "set to address"
"swap" is a 32 bit scratchpad variable.
The next two lines re-set the pointers to an address of state A that
is shifted the number of bytes dictated by the top two bits of the
6-bit change argument Tsel from above.
DW_A = (DWORD*) &STATE_A[Tsel>>4];
BT_A = &STATE_A[Tsel>>4];
Tsel = Tsel mod 16; "Reduce Tsel to a four bit (0-15) value."
"This shifts the values in A around according to Tsel, the Trans table and
the byte skew accomplished by the pointer re-assignment."
for (i=0, j=1; i<16; i++, j++) DW_A[Trans[Tsel] [ i ]] = DW_A[Trans[Tsel][ j ]];
"Shift around 4 bytes in state A."
swap = BT_A[3];
DW_A[0] = (DW_A[0]<<8) OR swap;
"Duplicate the first two 32 bit chunks at the array end."
DW_A[16] = DW_A[0]; DW_A[17] = DW_A[1];
That's really all there is to it. There are some little nits that
are fleshed out in the text but this is the crazy notion I am
proposing.
Please, If I have missed the target again, let me know how
I might better answer.
Thanks
Paul
------------------------------
From: Bryan Olson <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
Date: Mon, 19 Feb 2001 17:25:16 -0800
Paul Lutus:
> jtnews:
[...]
> > Let me clarify what I mean by sequentially enumerate.
> > By sequentially enumerate I mean successively enumerate
> > all possible transcendental numbers starting from zero
> > to infinity.
> >
> > It doesn't matter that the set itself is infinite.
>
> But it does, by your own definition. If the set is infinite,
> then enumerating "all possible transcendental numbers" is not
> an achievable goal.
He's allowing algorithms that run forever.
The answer is, as others have pointed out, "no". The
transcendentals are uncountable.
>
> > My question is analagous to:
> >
> > Is there an algorithm to sequentially enumerate all
> > prime numbers?
> >
> > Here the answer is yes.
>
> NO, the answer is "no." The problem is your use of the term
> "all possible prime numbers." The set is infinite, therefore
> there is no way to assign a meaning to "all."
Sure there is. If there exists a prime such that the
algorithm will never reach it, then the algorithm does
not enumerate all primes. If for every prime p, the
algorithm eventually reaches p, then it enumerates all
the primes.
--Bryan
------------------------------
From: Bryan Olson <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
Date: Mon, 19 Feb 2001 17:25:48 -0800
Paul Lutus:
> jtnews:
[...]
> > Let me clarify what I mean by sequentially enumerate.
> > By sequentially enumerate I mean successively enumerate
> > all possible transcendental numbers starting from zero
> > to infinity.
> >
> > It doesn't matter that the set itself is infinite.
>
> But it does, by your own definition. If the set is infinite,
> then enumerating "all possible transcendental numbers" is not
> an achievable goal.
He's allowing algorithms that run forever.
The answer is, as others have pointed out, "no". The
transcendentals are uncountable.
>
> > My question is analagous to:
> >
> > Is there an algorithm to sequentially enumerate all
> > prime numbers?
> >
> > Here the answer is yes.
>
> NO, the answer is "no." The problem is your use of the term
> "all possible prime numbers." The set is infinite, therefore
> there is no way to assign a meaning to "all."
Sure there is. If there exists a prime such that the
algorithm will never reach it, then the algorithm does
not enumerate all primes. If for every prime p, the
algorithm eventually reaches p, then it enumerates all
the primes.
--Bryan
------------------------------
From: [EMAIL PROTECTED] (John Savard)
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b.Can I ALWAYS find a transcendental
numberbetween a and b?
Date: Tue, 20 Feb 2001 01:10:38 GMT
On Mon, 19 Feb 2001 21:22:55 +0100, Jan Kristian Haugland
<[EMAIL PROTECTED]> wrote, in part:
>This, however, doesn't show that the set of
>transcendentals is uncountable (see other comments
>about the rationals).
That may not show it, but indeed the set of transcendentals _is_
uncountable.
The reals are uncountable, by Cantor's diagonal proof.
A transcendental is any real that is not algebraic.
The algebraic numbers are countable. (Proof: there exists a Godel
numbering of algebraic numbers.)
The union of two countable sets is countable. (An uncountable set is a
set which is not countable.)
Therefore, since the reals, which are uncountable, are the union of
the algebraic numbers, which are countable, and the transcendental
numbers, the transcendental numbers are uncountable.
John Savard
http://home.ecn.ab.ca/~jsavard/crypto.htm
------------------------------
From: [EMAIL PROTECTED] (John Savard)
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b?
Date: Tue, 20 Feb 2001 01:06:22 GMT
On Mon, 19 Feb 2001 15:01:18 -0500, jtnews <[EMAIL PROTECTED]>
wrote, in part:
>Can I ALWAYS find a transcendental number
>between a and b?
Yes.
The algebraic numbers are nowhere dense.
John Savard
http://home.ecn.ab.ca/~jsavard/crypto.htm
------------------------------
From: [EMAIL PROTECTED] (John Savard)
Crossposted-To: sci.math
Subject: Re: Given any arbitrary numbers a and b. Can I ALWAYS find a transcendental
number between a and b?
Date: Tue, 20 Feb 2001 01:17:47 GMT
On Mon, 19 Feb 2001 16:54:34 -0500, jtnews <[EMAIL PROTECTED]>
wrote, in part:
>However, I'm not sure what uncountable
>and countable mean. I'm going to have
>to lookup these terms myself.
Although my web page is mostly about cryptography, at
http://plaza.powersurfr.com/jsavard/other/infint.htm
you will find a good explanation of this subject.
John Savard
http://home.ecn.ab.ca/~jsavard/crypto.htm
------------------------------
From: [EMAIL PROTECTED] (John Savard)
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Tue, 20 Feb 2001 01:14:56 GMT
On Mon, 19 Feb 2001 11:37:47 -0500, jtnews <[EMAIL PROTECTED]>
wrote, in part:
>Let me clarify what I mean by sequentially enumerate.
>By sequentially enumerate I mean successively enumerate
>all possible transcendental numbers starting from zero
>to infinity.
>It doesn't matter that the set itself is infinite.
>My question is analagous to:
>Is there an algorithm to sequentially enumerate all
>prime numbers?
>Here the answer is yes. Even though the set of all
>prime numbers is infinite, there is an algorithm to
>enumerate the entire set starting from zero to infinity.
Since the set of transcendental numbers is not merely infinite, but
also uncountable, they cannot be placed in one-to-one correspondence
with the integers.
John Savard
http://home.ecn.ab.ca/~jsavard/crypto.htm
------------------------------
From: [EMAIL PROTECTED] (John Savard)
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Tue, 20 Feb 2001 01:16:07 GMT
On Mon, 19 Feb 2001 19:56:40 GMT, Steve Leibel <[EMAIL PROTECTED]>
wrote, in part:
>What about the rationals? Between any two there are infinitely many
>other rationals, yet we can easily generate all the rationals using an
>algorithm -- just follow the arrows in the standard demonstration that
>the rationals are countable.
You are correct. However, the transcendentals are uncountable, because
the reals are uncountable (by Cantor's diagonal proof) and the
algebraics are countable.
John Savard
http://home.ecn.ab.ca/~jsavard/crypto.htm
------------------------------
From: "Dik T. Winter" <[EMAIL PROTECTED]>
Crossposted-To: sci.math
Subject: Re: Is there an algorithm to sequentially enumerate all transcendental
numbers?
Date: Tue, 20 Feb 2001 01:35:53 GMT
In article <%cck6.343$[EMAIL PROTECTED]> "Paul Lutus"
<[EMAIL PROTECTED]> writes:
> But it does, by your own definition. If the set is infinite, then
> enumerating "all possible transcendental numbers" is not an achievable goal.
Oh, it is not, but not for the reason you supply.
> NO, the answer is "no." The problem is your use of the term "all possible
> prime numbers." The set is infinite, therefore there is no way to assign a
> meaning to "all."
But there is. {p | p in N, p prime} is the set of all prime numbers.
There is a definite meaning of the word all here, namely there are no
prime numbers that are not in the given set.
>
> When you speak of an algorithm to enumerate "all possible" members of a set,
> the implication is that the algorithm will create the list and stop. If the
> set is infinite, this is not possible.
Why do you think the algorithm should stop? Moreover, why do you think
"enumerate" should implie "generate"? Given the infinite set
{n | n in N, n square} you say there is no algorithm that "enumerates"
all elements of that set, nevertheless, the algorithm (function)
f: i (in N) -> i^2, is such an enumeration, it assigns an integer
number to every element of the set.
> I think the problem is in how you are describing the problem, not the
> problem itself. You want to list an arbitrarily large number of
> transcendental numbers, not "all possible."
But that was indeed *not* what was asked. The question was about
*all possible*, and there is no enumeration possible in the case of
transcendental numbers. There is for rational numbers, algebraic
numbers, and even for constructable numbers; but in the latter case
the enumeration is not constructable. Confusing isn't it? But that
is mathematics.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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