December 2003 - cypherpunks-legacy

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

17 Dec '03

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 where m and n -> infinity. Another way of saying this is that there are Reals for which membership in a set of Reals, because they are uncountable and therefore unrepresentable, is not possible. From a set perspective this means, R = [[-infinity, ..., m] [n, ..., 0] [0, ..., p] [q, ..., +infinity]] such that there are uncountable numbers between m,n or p,q; etc. Clearly in contradiction with the base axioms of mathematics as described by Euclid in defining a line. > 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. There are three ways of looking at the total of mathematics, symbolic set theory (Bourbaki School) geometric (Euclidian) While it is true the 3 are equivalent, the choice of approach does have a relevance on how difficult, if even tractible, the proof is for any particular concept. > It is often convenient, such as when drawing contour maps, to > consider the complex numbers to be in 1-1 correspondence with the > points of the plane. However, I wouldn't necessarily consider > regions of the complex plane to have "area" in the Euclidian > sense. The Euclidian plane IS a complex plane. The -j or -i used in Complex symboligy simply means 'rotate the second axis of measure 90 degree counter-clockwise (by agreement) to the axis of the first measure'. If it requires 2 or more numbers (integer or real) taken as a set to represent the quantity it is a Complex. Regions of a Complex plane are directly comparable (1-to-1) with the concept of 'area' on the Euclidian plane. > We can't physically draw a line segment to arbitrary high > precision. Irrelevant. If the hole in my approach is that I can't draw a line of arbitrary precision in practice then your own 'uncountable numbers' argument falls for the same reason because if it is truly uncountable you can't point to it on a number line and say "there is an uncountable". This discussion is one of principles, not one of practice. > We can conceive of the notion of line segments being > in 1-1 correspondence with the reals, but we can specify at most > countably many "finitely denumerable" line segments if we wish to > discuss their lengths individually. To say there are Reals for which there is no linear representation is the same as saying there are lengths which can't be measured. Now since a line is nothing but a set of points, which don't have size, but only position this obviously holds no water, unless you are saying it is not possible to place two points arbitrarily close together, which would imply that points have some sort of width, clearly against the definition of a point. This all goes back to what I said in a earlier post, the problem comes from our axiomatic (ie taken on faith, unprovable) use of infinity. Without a clear and precise dilineation of those axioms prior to the proof such conclusions are worthless. Several of you have said "infinity is not a number", this is an axiom. Change the axioms and the whole structure changes. I am simply saying that perhaps we should look at the "infinity is not a number" axiom, much as geometers look at Euclids Fifth Postulate. There is nothing inherent in nature that prefers one axiomatic expression of infinity over the other. By changing our axiomatic definition of infinity we reduce the sets we have to work with from [Integer, Irrational, Real, Complex] to [Integer, Real, Complex]. Now, whether it is worth the trouble is at this time unanswerable because nobody has ever done the research. Jim Choate CyberTects ravage(a)ssz.com "The laws of mathematics, as far as they refer to reality, are not certain, and as far as they are certain, do not refer to reality." Albert Einstein

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Debt (calling) Cards Re: Newt's phone calls
by Sean Roach 17 Dec '03

17 Dec '03

At 04:58 PM 1/17/97 -0800, Rich Graves wrote: >What does concern me is personal privacy. I don't want people tracking >my movements or purchasing patterns, so I prefer to use cash or >pseudonymous debit cards. This reminds me, I got a sprint 5 minute sample calling card when I got my college texts. I suspect that other people on this list got one as well. Mine is completely unused and although transferring the actual cards would cost, (the stamp and envelope), would anyone be interested in trading the numbers? This one is registered as being handed out in Weatherford Okla. Five minutes of long distance is trivial, so I would be willing to trade in regular e-mail, as opposed to through a middleman.

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Re: Airport security [no such thing]
by Sean Roach 17 Dec '03

17 Dec '03

At 08:04 PM 1/17/97 EST, Dr.Dimitri Vulis wrote: >snow <snow(a)smoke.suba.com> writes: > >> > I say we make people fly naked and ship their stuff via UPS ;). >> >> >>shudder<< >> >> No!. A clear case of the cure being worse than the disease... > >Why. it might be fun - those long flight can be so boring with clothes on... > Wouldn't work. A sworn member of the holy order of the restitution for the crimes of my enemies, has an explosive device buried within h[is/er] bowels. When (s)he goes up, alone, (s)he will take 100+ enemies with h[im/er]. Of course (s)he has to be willing to die, or be duped into thinking that (s)he really has appendicitis. So a courier would still be easy to find. Or, a cancer infected member of said religious cult, knowing that (s)he will soon die anyway, and probably atone for h[is/er] many sins, ((s)he hasn't really done anything rightous, like kill one of the enemy), chooses to have h[is/er] cancerous bowels, up to the stomache, removed, and replaced with a bomb. (s)he then heals up a little while, being fed intraveinously, until the time of h[is/er] atonement. A makeup artist erases the signs of disease that (s)he carries and a large dose of pain killer is given to h[im/er] to take right before entering the airport. (s)he dies knowing that (s)he is going straight past heaven to an even higher level.

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Re: Newt's phone calls
by Sean Roach 17 Dec '03

17 Dec '03

At 03:44 PM 1/17/97 -0600, Igor Chudov wrote: >It is my understanding that an ordinary banking consumer that >uses "pay bills by phone" service cannot transfer $$ to any arbitrary >account. There is a limited list of permitted accounts, such as utility >companies. Therefore, the risk of unauthorized transfer is very >limited. It is limited, both accounts have to have the same password, (four character, numeric.) If I want to transfer funds from another persons account to one that I control, all I have to do is change the password on one of them to that of the other, transfer funds, and change the password back. By the time that my victim found out, (via the little letter "per your request, $xxx was transferred to accnt#123456 from accnt#7890"), I would be long gone.

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Re: 'Monster' numbers
by winsock＠rigel.cyberpass.net 17 Dec '03

17 Dec '03

At 5:08 PM 1/19/1997, Jim Choate wrote: >In reference to numbers which you can't describe, if you examine the work >they are ALL in the Complex domain, none of them are Real's. This is incorrect. The argument outlined applied to real numbers only. If you wish to refute the argument please do so. You might find it helpful to review the reference in Smullyan's book. >If there existed a Real for which we could not describe this would imply >that we could not draw a line of that length. The subtlety of the idea lies in the fact that you can't make a statement like "$foo is not describable" where $foo means anything. Any number for which you can say, "I can draw a line which is $bar units long" is describable. Math Man

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Re: infinity & set membership
by nobody＠huge.cajones.com 17 Dec '03

17 Dec '03

At 4:36 PM 1/19/1997, Jim Choate wrote: >"The least upper bound of a set of real numbers is often called the >supremum, the greatest lower bound its infimum. In general the supremum and >infimum of a set ARE MEMBERS OF THE SET or at least LIMITS OF SEQUENCES OF >MEMBERS OF THE SET." > >[capitalization is mine] > >In this case the suprenum is infinity. > >Introduction to Calculus and Analysis >Courant and John >Vol. 1, pp. 97, Section e. >1965 Edition >Library of Congress: 65-16403 I presume this is a reference to Jim's earlier post: At 1:19 PM 1/19/1997, Jim Choate wrote: >Forwarded message: > >> Date: Sun, 19 Jan 1997 12:20:45 -0600 >> From: Sir Robin of Locksley <tozser(a)stolaf.edu> > >> >Is it possible to prove that number 0.1234567891011121314151617181920... >> >iz irrational? >> >> Most definately. All you need to do is prove that the set of this number is >> uncountable, ergo is irrational. If you have friend who have done math in real >> analysis they can explain more. > >irrational means 'not expressible as the ratio of two integers', this does >not imply uncountable. > >Whether 0.12345678910111213... is irrational or not depends on how we choose >to define a rational number where the denominator and numerator are both >approaching infinity and how quickly those approaches occur. > >[ What is your definition of infinity/infinity ?] > >As alluded to before, > > 0.12345678910111213 = 12345678910111213.... / n > > (where n = infinity) > >0.123456789101112... is certainly countable because by the definition of >countable it must be 1-to-1 with the counting numbers (ie non-negative >integers), which this one clearly is since it contains each positive >non-zero integer (ie the set of numbers required to produce the number are >clearly less, 1 less to be exact, than the counting numbers). Of course this same argument could be used to claim that pi is also rational. pi = 314159.... / n (where n=infinity). What could be the problem here? One problem might be that an integer (not including infinity) divided by "infinity" should be 0. The p/q definition of rational numbers does not rely on "infinity" being a member of the set of integers. Also, we rarely hear this dialogue: "Give me an integer." "Okay - infinity!" Still, let's define a set which contains the integers as well as positive and negative "infinity". I will call this set the "Choate-integers." Let's add some reasonable rules for working with +inf and -inf. n is a non-zero positive traditional integer. a. -inf = +inf * -1 b. +inf * n = +inf c. +inf + n = +inf d. +inf / +inf = 1 Operations involving multiplication or division of +inf and zero are undefined. Using these definitions, you will find that the p/q definition of rational numbers operates as expected on the set of Choate-integers. The argument I supplied earlier for the rational nature of 0.1234... still holds. Math Man

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Take it to sci.math
by winsock＠rigel.cyberpass.net 17 Dec '03

17 Dec '03

Will all those people who are talking about set theory and the countability of various number systems take their discussion to sci.math, please? 'Cuz 1) All that stuff is irrelevant to crypto, its science or politics. 2) Not a single math comment made by anybody on any of those threads is even remotely correct. 3) You will recieve your proper chewing out on sci.math and 4) This bullshit is extremely annoying to those of us who has some real training in math, even more so than the regular bullshit on this list. Hey Sandy, you gonna censor these creeps once you start censoring?

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Re: Dedikend Cut's and such
by Kent Crispin 17 Dec '03

17 Dec '03

Jim Choate allegedly said: > > > Hi, > > 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. -- Kent Crispin "No reason to get excited", kent@songbird.com,kc@llnl.gov the thief he kindly spoke... PGP fingerprint: 5A 16 DA 04 31 33 40 1E 87 DA 29 02 97 A3 46 2F

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Re: Newt's phone calls
by Sean Roach 17 Dec '03

17 Dec '03

At 03:44 PM 1/17/97 -0600, Igor Chudov wrote: >It is my understanding that an ordinary banking consumer that >uses "pay bills by phone" service cannot transfer $$ to any arbitrary >account. There is a limited list of permitted accounts, such as utility >companies. Therefore, the risk of unauthorized transfer is very >limited. It is limited, both accounts have to have the same password, (four character, numeric.) If I want to transfer funds from another persons account to one that I control, all I have to do is change the password on one of them to that of the other, transfer funds, and change the password back. By the time that my victim found out, (via the little letter "per your request, $xxx was transferred to accnt#123456 from accnt#7890"), I would be long gone.

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Re: 'Monster' numbers
by WinSock Remailer 17 Dec '03

17 Dec '03

At 5:08 PM 1/19/1997, Jim Choate wrote: >In reference to numbers which you can't describe, if you examine the work >they are ALL in the Complex domain, none of them are Real's. This is incorrect. The argument outlined applied to real numbers only. If you wish to refute the argument please do so. You might find it helpful to review the reference in Smullyan's book. >If there existed a Real for which we could not describe this would imply >that we could not draw a line of that length. The subtlety of the idea lies in the fact that you can't make a statement like "$foo is not describable" where $foo means anything. Any number for which you can say, "I can draw a line which is $bar units long" is describable. Math Man

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