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Re: I beg you, PLEASE prove that 0.123456789101112131415 is IRRATIONAL
by lucifer＠dhp.com 17 Dec '03

17 Dec '03

At 12:20 PM 1/19/1997, Sir Robin of Locksley wrote: >>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. The set of this number contains one member - the number. Therefore, this is a countable set. A rational number is a number which can be described as a fraction in which the numerator and the denominator are integers. Jim Choate implied that "infinity" is an integer. This is not correct. The set of integers is said to be infinite. Infiniteness is a property of the set, not a member of the set. All rational numbers are expressible in a decimal form of finite length or in a decimal form which ends in a repeatable string of digits. (Consider the process by which each successive digit is found. It is a fraction of the denominator and some remainder with a zero added. The remainder is less than the denominator multiplied by the base, 10 in this case. When a remainder repeats, the state of the algorithm repeats, and its output will therefore be repeated. Since the remainder is bounded, the algorithm must repeat a state. Note that this holds regardless of the base used.) If our number, 0.1234..., does not repeat then it is not rational. Let's show that it does not repeat. If it repeats, there is a sequence of digits which repeats indefinitely. Such a sequence must be longer than the longest integer. (Because if it was part of an integer, the other parts of the integer would not always be the same and it couldn't repeat.) But, there is no longest integer. Therefore there is no repeating sequence. Therefore our number, 0.1234..., is not a rational number. Math Man

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KeyKOS related system lecture at Stanford
by Bill Frantz 17 Dec '03

17 Dec '03

I received the following from Jonathan. Those of you who are interested in KeyKOS should know that his system, EROS, is an architectural descendent of KeyKOS. --------------------------------------------- Date: Sun, 19 Jan 1997 15:13:11 -0500 From: "Jonathan S. Shapiro" <shap(a)eros.cis.upenn.edu> To: ... Subject: Talk info NOTE TIME CORRECTION I'll be giving a talk on Wednesday at Stanford in the EE380 seminar. Details on the seminar can be found at http://www-leland.stanford.edu/class/ee380/ under 'winter 1996'. The location and time are: Wednesday at 4:15pm Hewlett Packard Auditorium (Room B1) Gates Computer Science Building Stanford Univeristy The basic structure of the talk will be: explain what active networking means, and why confinement and fault isolation are required. (the problem) identify the properties needed for confinement and fault isolation proofs, what capabilities are, and why non-capability systems do not have these properties (why we need them) describe the objects named in EROS/KeyKOS, and why these are the right choices for high performance (how to get performance) show how this system might be used to construct an active networking router or endpoint. (how to use them for this problem) I'm still working on the talk, so the order may get rearranged. Hope to some or all of you there. Jonathan

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Igor@home = Secret Squirrel
by Jim Choate 17 Dec '03

17 Dec '03

Hi all, I am motivated to post this, something I would normaly not do, because of the issue of trust of moderators that has arisen lately. Igor and myself were in a discussion about number theory last nite. We reached a point of impass. Igor made the suggestion that we bring the discussion to cpunks. I requested that we drop the issue and specificaly not bring it to cpunks. My reasoning had several facets. First, the position that I was in is probably wrong. My motivation was to examine the methodology of dis-proof (which I was disappointed with). Also, it is not a serious issue with me and whether my particular view is right or wrong is irrelevant. Cypherpunks is not the place for truly technical discussions (as is apparent with even a cursory glance). Traffic on this list practicaly drowns my ability to filter it and more drivel I (and probably you) don't need. And finaly, the discussion was fine for a Saturday afternoon discussion, I don't have the interest or time to carry it out over days or weeks, I have to work for a living. Igor agreed to drop the issue and not to bring it to the list. However, this morning I find Secret Squirrel posting some questions which are taken practicaly verbatin from my exchange w/ Igor. Seemed like a truly improbable event. Take it for what it's worth. "An appeal to authority is an appeal to a human being just like yourself who does not have your best interests at heart. What makes you believe they are able to resolve your problems better than yourself?" Jim Choate CyberTects ravage(a)ssz.com

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'Monster' numbers
by Jim Choate 17 Dec '03

17 Dec '03

Hi, 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. Complex numbers deal with areas, not with lengths. If there existed a Real for which we could not describe this would imply that we could not draw a line of that length. Something which is clearly contrary to the axiomatic assumptions of lines and their construction (ie points have no dimension, only position, and lines are infinite sequences of points). A clear Complex example of a 'monster' is Sierpenski's Gasket. It fills an area but has no measurable surface area. Jim Choate CyberTects ravage(a)ssz.com

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[Math Noise]
by mpd＠netcom.com 17 Dec '03

17 Dec '03

Jim Choate <ravage(a)EINSTEIN.ssz.com> writes: > 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. 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 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. > Complex numbers deal with areas, not with lengths. 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. > If there existed a Real for which we could not describe > this would imply that we could not draw a line of that > length. We can't physically draw a line segment to arbitrary high precision. 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. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd(a)netcom.com $ via Finger. $

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'Making Money Fast"
by cyberdude＠iname.com 17 Dec '03

17 Dec '03

I took YOUR address from this news group, because it's obvious that you are aggressive and " WANT TO MAKE MONEY FAST ' For more information contact cyberdude(a)iname.com

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THIS IS THE LAST UNMODERATED MESSAGE TO CYPHERPUNKS
by John Gilmore 17 Dec '03

17 Dec '03

OK, I think we're finally ready to go live with a moderated cypherpunks list (modulo glitches). Operators (well, Sandy) are standing by! John Gilmore

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Re: Numbers we cannot talk about
by ichudov＠algebra.com 17 Dec '03

17 Dec '03

Paul Elliott wrote: > -----BEGIN PGP SIGNED MESSAGE----- > > At 10:48 PM 1/18/1997, Secret Squirrel wrote: > > >Is it REALLY true that there are real numbers that cannot be > > >generated by any algorithm? Some guy said that since the set of > > >algorithms is countable, but the set of real numbers is more than > > >countable, there must be some numbers for which there is no > > >algorithms that generate them. > > > > There are sets of real numbers whose existence we can prove, but which > > we cannot otherwise describe. This is more extreme than being > > "generated by an algorithm". We can't even tell somebody which > > numbers to generate! (I take "to generate" here to mean "to compute a > > decimal approximation.") > > > > The set of real numbers is uncountable as is the set of subsets of the > > real numbers. Yet, we have only countably infinite ways to describe > > sets of numbers. > > > > All sets of numbers which we can describe can be described with a > > finite set of symbols. (Human beings are unable to distinguish > > between an infinite number of states.) The set of combinations of > > this finite set is infinite, but countable. > > > > Perhaps the axioms in set theory that tells us that the integers > have an uncountable number of subsets is, in point of fact, false. > Perhaps only those subsets of the integers that can be described > by an algorithm exist (actually, contrary to what the usual axioms of > set theory assert). It is very interesting. My limited understanding of this approach is that they say that only things that can be constructed by some positive method exist (please correct me if I am mistaken). But the question is, where do they stop and what exactly is "construction?" Say, does sqrt(2) "exist" in their sense of the world? We know we can calculate any given number of digits in it, is that enough? > We know that the set of axioms which tell us that there are unaccountably > many reals can be satisfied by a countable model! > (Downward Louwenheim Skolem Tarski theorem.) > > I know that Standard mathematical axioms yields lots of interesting > results, but when it talks of the infinite and we are dealing > with a practical subject like cryptography or even physics it > should not be taken too seriously. (With respect to uncountable sets.) Some of the applications of these theories are very relevant. For example, a theorem that proves that it is impossible to write a program that would determine if any other program would stop or loop forever, is very relevant and interesting. - Igor.

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THIS IS THE FIRST MESSAGE TO THE MODERATED CYPHERPUNKS LIST
by John Gilmore 17 Dec '03

17 Dec '03

If you see this, Sandy has approved it and forwarded it to the list. Of course, this is modulo glitches and bugs. It may take a few days to get everything 100% right. And the messages will undoubtedly arrive on your site in a different order than the order in which they were sent (due to queueing at various places, mostly on toad.com) Welcome to (what I hope is) a more useful and less confrontative cypherpunks list. Now, can we get back to talking about cryptography? John Gilmore

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[Math Noise]
by mpd＠netcom.com 17 Dec '03

17 Dec '03

Jim Choate <ravage(a)EINSTEIN.ssz.com> writes: >> One can construct the reals from the rationals quite easily >> using any of several well-known methods, such as >> equivalence classes of Cauchy sequences, or Dedikind cuts. > So you are saying that the Reals are a subset (ie can be > constructed from) of the Rationals? Yes, the Reals can be constructed from the Rationals. No, the Reals are not a subset of the Rationals. Fortunately, construction in mathematics is not simply limited to the taking of subsets. The Rationals can be constructed from the Integers, for instance, by multiplication and the taking of appropriate equivalence classes. In fact, everything can be built out of two sets and the axioms of Set Theory. > I can create a number which is not representable by the > ratio of two integers from two numbers which are > representable by ratios of two integers? Er, no. But you can create a number which is not representable as a fraction as a limit point of very many fractions. > That's a nifty trick indeed, I am really impressed. Thank-you. :) > Cauchy produced a test for testing convergence. I fail to > see the relevance here, but please expound... Cauchy sequences are useful for adding limit points to a set of things because their convergence criteria is very simple, and it is conceptually easy to take all Cauchy sequences whose elements come from a given set. If we then consider equivalence classes of those Cauchy sequences which converge to the same limit, and consider an element of the original set to correspond to the class containing the sequence all of whose members are that element, we can consider the classes to form a "completion" of the original set by addition of all its limit points. Similarly, the Reals are the completion of the Rationals. > Dedekind Cut: > "Thus a nested sequence of rational intervals give rise to > a seperation of all rational numbers into three classes." > Just exactly where does this allow us to create Reals? Finite ordered sets have maximum elements. Bounded infinite ordered sets have Least Upper Bounds, which may be a limit point as opposed to being an actual member of the set. 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. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd(a)netcom.com $ via Finger. $

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