December 2003 - cypherpunks-legacy

Re: [Math Noise] (fwd)
by Paul Foley 17 Dec '03

17 Dec '03

On Sun, 19 Jan 1997 22:34:05 -0600 (CST), Jim Choate wrote: Forwarded message: > Yes, the Reals can be constructed from the Rationals. No, the > Reals are not a subset of the Rationals. An arbitrary Real can be constructed from the Rationals. If we accept the proposition, as posed apparently by you and others, of uncountable Reals then your 'assumption' fails, otherwise the 'uncountable' members would be countable. There are no "uncountable" numbers -- uncountability is a property of the set, not an individual member of the set. The set of reals cannot be placed in one-to-one correspondence with the integers (strictly, positive integers, but it amounts to the same thing). > Dedekind Cuts are a simple abstraction, often used to construct > the Reals from the Rationals in undergraduate calculus courses. > Conceptually, one makes a single "cut" in the set of Rationals, > dividing it into two parts, all of the members of one part being > greater than all of the members of the other. The number of ways > of doing this correspond to the Reals. The number of cuts are 1-to-1 with the Reals, they are not the Reals. There is no way I can make a cut which is 3.1527, only 1-to-1 with the number(s). Important distinction. Cut the rational numbers into two sets, A containing all the negative rationals and all those that have squares less than 2, and B containing all the positive rationals that have squares greater than 2. There you have a cut which is 1.41421... (i.e., sqrt(2)). So you can define irrational numbers from the rationals (an irrational number is a cut such that the first set (A) has no largest member and the second set (B) has no smallest member). -- Paul Foley <mycroft(a)actrix.gen.nz> --- PGPmail preferred PGP key ID 0x1CA3386D available from keyservers fingerprint = 4A 76 83 D8 99 BC ED 33 C5 02 81 C9 BF 7A 91 E8 ---------------------------------------------------------------------- Sometimes a feeling is all we humans have to go on. -- Kirk, "A Taste of Armageddon", stardate 3193.9

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Re: [Math Noise] (fwd)
by ichudov＠algebra.com 17 Dec '03

17 Dec '03

Jim Choate wrote: > Forwarded message: > > > Only countably many real numbers, or members of any uncountable > > set, are denumerable. It is the property of being uncountable, > > rather than of being real or complex, which is important here. > > In short you are saying there are Reals which can not be expressed in the > format: > > AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1 All reals are equivalent to sequences of digits, but there are reals such that there is no algorithm to generate their digits. It happens because there are "more" real numbers than algorithms. > > In general, only countably many members of any uncountable set > > can be precisely specified within any formal system, given names > > comprised of strings of symbols, or other similar things. > > And I contend that ANY number which is Real can be expressed by the decimal > expansion above. Which clearly qualifies as a formal system. I suggest the following mental exercise. FORGET FOR A MOMENT ABOUT REAL NUMBERS. Let's deal with mummies: DEFINITION: I define a mummy as possibly infinite sequence of characters, separated by one dot, such that only characters abcdefghij are allowed. Also, mummies that are represented by finite sequences of characters are by this definition equivalent to mummies that end with an infinite sequence of letters "a". END DEFINITION. Examples: dce.abdefhaabdaaa ae.cacacacacacaca... and so on. Obviously, some of the mummies, such as c.cccccc... (with an ininite sequence of "c") CAN be generated by algorithms. The interesting fact, that i will prove below, is that some of them cannot be generated by any algorithm. THEOREM: The set of mummies is more than countable PROOF: if it is countable, we can construct a mummy that is not counted. it is easy. THEOREM: there are mummies such that there is no algorithm that can print them. PROOF: the set of mummies is more than countable, the set of algorithms is countable, therefore there is no way to construct a one-to-one correspondence between mummies and algorithms. Do you agree? Now let's back to the original problem of real numbers: the only difference between mummies and real numbers is that digits 0123456789 are replaced by characters abcdefghij. Not a whole lot of difference, so everything that applies to mummies applies to real numbers. - Igor.

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Re: GSM technology
by Adam Back 17 Dec '03

17 Dec '03

Steve Shear <azur(a)netcom.com> writes: > [very useful explanation of GSM TDMA, and the new HSCSD] 64 kbit/s tied to ISDN is very nice technology. > ENCRYPTED GSM LINKS > If the GSM phone includes a data port, as some already do, just connect > your laptop, dial your ISP and 'push' the encrypted traffic over that link > (e.g., using TCP/IP and PGPfone). This would obviously be possible, but to my mind reduces the appeal of the system. Not every one has a laptop. Even if the potential user does have a laptop, booting windows95, and starting up PGPfone is an onerous task compared to just dialing a number on a mobile phone. It's a user friendliness issue, and a question of ergonomics. A mobile phone is more portable than a GSM phone with a laptop plugged into the data port. You can't fit the laptop and GSM phone combination into your shirt pocket. What are you going to use for a handset? Radio operators headphone (with mike attached to the headset) plugged into the laptop? That's more dangling wires, and makes the system less portable, and even more onerous to setup (take laptop from carry bag, plug in headphones, plug in phone data port, wait for laptop to boot, etc). Even for crypto enthousiasts, I would submit that many would neglect to go through the hassle of going through PGPfone for most conversations, and would instead just use the mobile phone in the clear (or with A5 encryption). This for similar reasons to the situation with PGP itself, many people rarely use PGP, even though there is abundant software available to use it seamlessly with most mail readers. (I can vouch for mailcrypt.el the emacs interface to PGP, and use it for to anyone who has a PGP key, and does not express displeasure at receiving encrypted email). Also, the cell phone tarriffs may be higher if you need higher bandwidth to get the software only voice codec implementations in PGPfone to produce equivalent full-duplex voice quality to that expected from a digital mobile phone. Not knocking the mobile phone and laptop combination for the purpose of having mobile TCP/IP access from a laptop, I know several people who have this combination, though only at 9.6kbits, and even at that speed it is very neat. Adam -- print pack"C*",split/\D+/,`echo "16iII*o\U@{$/=$z;[(pop,pop,unpack"H*",<> )]}\EsMsKsN0[lN*1lK[d2%Sa2/d0<X+d*lMLa^*lN%0]dsXx++lMlN/dsM0<J]dsJxp"|dc`

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Re: [Math Noise] (fwd)
by Mike Duvos 17 Dec '03

17 Dec '03

Jim Choate <ravage(a)EINSTEIN.ssz.com> writes: > An arbitrary Real can be constructed from the Rationals. If > we accept the proposition, as posed apparently by you and > others, of uncountable Reals then your 'assumption' fails, > otherwise the 'uncountable' members would be countable. We can construct the Reals from the Rationals without having to speak of each specific Real while doing so. A formal system, having only a countable number of strings of symbols from its alphabet, can speak of "The Real Numbers" even though it cannot speak of "The Real Number X" for every single X in the Reals. > A 'limit point' is not the same as 'equal to'. Arbitrarily > close is not equivalent, inherent in the definition of a > limit point is the concept of 'little o' and 'big o', or > worded differently our axiomatic definition of infinity. In standard analysis, the limit of a sequence A[n] is a value x such that given any positive epsilon, no matter how small, we can find a point in the sequence such that all its members after that point are within epsilon of x. Such a limit, if it exists, is unique and exactly defined. "Little o" and "big o" are concepts from complexity theory and I am not precisely sure why you feel they need to be mentioned. > I can, but there is no fundamental rule in mathematics > that requires me to ignore the small but distinct > difference between the element in the original set and the > sequences used to approximate it. In the general case, the limit point will not be a member of the sequence which approximates it. Although every member of the sequence is a finite distance away from the limit, the limit itself is, as I previously mentioned, exactly known without any ambiguity. > It is something that must be agreed upon by accepting a > particular axiomatic definition of infinity. An "infinity" is simply the property of being able to be put in 1-1 correspondence with a proper subset of oneself. > The number of cuts are 1-to-1 with the Reals, they are not > the Reals. There is no way I can make a cut which is > 3.1527, only 1-to-1 with the number(s). Important > distinction. Mathematical objects are sets with structure. We generally consider two mathematical objects equivalent if there exists a 1-1 correspondence between the respective sets which is structure preserving. What the actual members of the set are, and how they were constructed, is usually unimportant. For all practical purposes we may refer to any mathematical object isomorphic to the Reals as "The Reals", without any confusion. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd(a)netcom.com $ via Finger. $

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how to get off this list
by Marc 17 Dec '03

17 Dec '03

How on earth an i meant to get off this mailing list??? have tried loadz of the un scribe says its undone it but it bloody hasnt someone help me

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Re: Thoughts re moderation, filtering, and name changes
by hjk 17 Dec '03

17 Dec '03

Just one thing about the names of the different lists: I received a mail listing the changes to come. It was very easy to understand that I am going to get the moderated list if I do not change my subscription.I'll find out if I will miss something. We'll see. I wonder who is so brainless, not to understand what's going on. Do you really think the subscribers are pure Idiots? Heinz-Juergen Keller email: hjk(a)ddorf.rhein-ruhr.de

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Re: GSM technology
by Nurdane Oksas 17 Dec '03

17 Dec '03

On Fri, 17 Jan 1997, Adam Back wrote: > > Steve Shear <azur(a)netcom.com> writes: > > [very useful explanation of GSM TDMA, and the new HSCSD] > > 64 kbit/s tied to ISDN is very nice technology. Does anyone use ADSL lines? or are they still very expensive? > > ENCRYPTED GSM LINKS > > If the GSM phone includes a data port, as some already do, just connect > > your laptop, dial your ISP and 'push' the encrypted traffic over that link > > (e.g., using TCP/IP and PGPfone). [...]

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Re: [Math Noise] (fwd)
by Jim Choate 17 Dec '03

17 Dec '03

> > And I contend that ANY number which is Real can be expressed by the decimal > > expansion above. Which clearly qualifies as a formal system. > > To be a formal system of the type required, you would also have to > specify deterministic rules that could generate the "Ai" values. The > key distinction is between "expressed by" and "generated by". But we do have a formal system for generating those terms. Dedekind Cuts used to enumerate the value of an arbitrary length compared to the unit length.

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Re: One time pads and randomness?
by Rick Osborne 17 Dec '03

17 Dec '03

>[Blurb on OTPs ...] I realize this has a limited amount of >messages before it is used up. But would this be secure? >Any suggestions, complaints, big gapping holes I missed? TMK, this is as close to a perfectly secure system as can be achieved. *AS LONG AS* the disks/discs are never compromised and never reused. As an interesting sidenote, if you read Tom Clancy's "Sum of All Fears" this is the type of system the CIA wants to implement even though the NSA doesn't want to back them because they think their (compromised) sytem is secure. Don't we all wish... _________ o s b o r n e @ g a t e w a y . g r u m m a n . c o m _________ "I thought your patience was infinte?" 'Since space and time are curved, the infinite sooner or later bends back upon itself and ends up where it began and so have I.'

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Re: Dedikend Cut's and such (fwd)
by Jim Choate 17 Dec '03

17 Dec '03

Forwarded message: > > Did a little research, Dedekind Cut's and such use Integers to define a > > SPECIFIC Real to a arbitrary BUT FIXED resolution. They do not define the > > set of Reals. > > They do. The set of all numbers representable by a DC is precisely > the set of reals. Then you are saying that using Dedekind Cut's it is possible to define the ENTIRE set of Reals? I am assuming that entire includes all those Reals which aren't representable by any algorithm and of which we can't even speak (even though we are). Another pretty nifty trick. Jim Choate CyberTects ravage(a)ssz.com

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