An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 19 Aug 2006 15:44 Virgil wrote: > In article <1155999688.567981.312170(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Dik T. Winter schrieb: >> >> >>> > Why then did you want to allow to assign finite sequences of edges only? >>> >>> To simplify matters. Natural numbers have in any base notation a finite >>> number of digits. In allowing finite sequences of digits I allow for >>> the occurrence of (for instance) both 0 and 00 and 000. So I give you >>> more freedom. >> But in fact that is less freedom because the sequence of edges per path >> is infinite. > > The sequence of edges leading up to any edge is finite. So a finite > sequence of 0's and 1's, 0 for left child, 1 for right child, uniquely > identifies each edge, but it takes an endless sequence of 0's and 1's to > identify any endless path. > > Thus in infinite binary trees, the set of edges is countable but the set > of endless paths is not. Ah yes, we've been over this. Since one extra path is created for every two edges added to the tree, there are clearly half as many paths as edges (floored, for a balanced tree, of course). It is unconscionably stupid to regard the number of paths as being "uncountable" and therefore greater than the "countable" number of edges, when clearly there are fewer paths than edges once the first three nodes are added to the root node, or binary point. The proof you have offered in the past is dishonest in the sense that it bijects the edges with the naturals using one interpretation of the tree, and then bijects the paths with the reals using another incompatible interpretation. Yes, we've been all through this. Thanks for the reminder. It's an example for my book. An example of what sucks with transfinite set theory. Have a nice day! ;) TO
From: Tony Orlow on 19 Aug 2006 15:47 Virgil wrote: > In article <1155998568.859179.65830(a)m79g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Franziska Neugebauer schrieb: >> >>> You have agreed to that this limit does not exist ("There is no L in >>> N"). So the sum is at least _not_ _finite_ (not in omega). >>> >> There is no L, nowhere. It is wrong to say that L is in |N and it is >> wrong to say that L is larger than any n e |N. Actual infinity does not >> exist! I told you that already several times. > > But "Mueckenh" has not been able to give us any compelling reason to > believe him, and we have compelling reasons not to. That's true! While he correctly illustrates that the infinite set of naturals cannot exist without infinite naturals, he fails to justify why this means the set is finite, and not that there exist infinite naturals. :) Tony
From: Tony Orlow on 19 Aug 2006 16:06 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: >>>> David R Tribble wrote: >>>>> mueckenh wrote: >>>>>> Learn: Infinitely many stairs require infinite height. But that it not >>>>>> admitted in mathematics. There have to be infinitely many stairs which >>>>>> give an infinitely long staircase ... >>>>> Yes, infinitely many finite-sized steps produce a staircase of infinite >>>>> height. That is admitted in mathematics. >>>> All too often "admitted" that those infinities are equal. >>>> >>>>>> ... but this staircase is allegedly of finite height. >>>>> No, the staircase is infinite in height, but each step is finite in >>>>> height. The distance between any two steps is finite, but the total >>>>> height of all the steps together (the entire staircase) is infinite. >>>>> >>>> So, the greatest possible value that any natural number can reach, in >>>> height, is infinite? >>> Hello Tony! >> Hi Brian!! How's tricks? >> >> I'll assume this is a question: >>> Q1: "Is the greatest possible value that any natural number can reach >>> infinite in >>> height?" >> Okay, you can rephrase it like that, sure. It had a question mark anyway. >> >>> I'll tell you the answer to Q1 when you tell me the answer to Q2... >> Okay. >> >>> You know that the sharpest corner of any triangle is less than 90 >>> degrees; the sharpest corner of an almost regular pentagon is more than >>> 90 degrees. >> Sure. >> >>> Q2: "Is the sharpest corner of a circle less than or greater than 90 >>> degrees?" >> So, if I answer this question, you will answer mine? Deal. >> >> A circle is a regular polygon with an infinite number of infinitesimal >> sides. The formula for the angle of the vertices of a regular polygon of >> n sides is 2*pi*(1/2-1/n). The limit of this formula as n->oo is pi. >> Each vertex of the circle has an angle of pi. That is, the change in >> angle at each vertex of the circle is infinitesimal, which makes it zero >> in the standard mathematical world. So, the angle at each vertex of the >> circle is greater than 90 degrees. >> >> Now, it's your turn. With the addition of each natural in turn, the >> width of the unary list is precisely equal at all times to its height, >> each time incrementing equal values as elements are enumerated, >> maintaining the squareness of the list, inductively demonstrable. So, >> how is it that you have such a list which has infinitely many >> successively greater columns [[shouldn't this be 'rows'?]], is square >> at every step, and yet, has no columns [[rows?]] with infinitely many 1's in it? > > The final drive shaft on our prewar Morris Minor rotated clockwise > (looking backwards). I've spent a lot of time wondering about the > direction of rotation of the drive shaft in a dog. Now I know. It > rotates in the chocolate direction. (It is incontrovertible that for > every drive shaft in every dog, there is no rotation in any direction > other than chocolate.) I hope you aren't feeding your dog chocolate. You do realize it's toxic for dogs, don't you, despite the doggy-chocolate craze of the '70's? Perhaps I should worry more about what YOU'RE swallowing, given the nature of the above statement. Should I call a doctor? You don't sound well. I'll take that as a joke, until I hear otherwise, and not worry. > > Well, it's the same, isn't it. When you arrive at this "list with > infinitely many rows, the bottom one is of length 42. There is no > bottom one of any length other than 42, and every row must have a > length, therefore it is 42. On the other hand, I admit it's rather easy > to prove that the bottom row in a bottomless list of rows is actually > infinite in length, green in colour, and weighs just 20g. I think you've read the Hitch-hiker's Guide enough times now. The answer is not 42. You have not properly formulated the question, Brian. Making fun of the process of formulation is not serving you well. Is this supposed to be the answer you promised? Can you even comment on the answer I gave you? Lay off the coffee. :| > > Oh, is that what you wanted? The last number in an unending sequence of > numbers is infinite? Sure. And pink. And yellow. And colourless. Sounds like something you've been drinking. Did it have little polka dot elephant ice cubes floating in it? > >> We made a deal, yes? Your turn. :) > > I note that your "circle" response makes pretty heavy use of undefined > terms (f-words and i-words), and relies on the properties of clearly > nonexistent objects. Well, good luck getting your paper accepted > somewhere. I beg your pardon? Please answer the following with yes or no: 1. Is the vertex angle of a regular polygon with n sides/vertices equal to 2*pi*(1/2-1/n), or pi-2*pi/n? 2. Is lim(n->oo: pi-2*pi/n)=pi? 3. Is there an undefined term or usage in the previous two statements? > > Brian Chandler > http://imaginatorium.org @ nothing.changed.i.c > >>> Certainly you claim that the width of this square, >>>> the count of all such values, is infinite. But are the values >>>> themselves, even potentially, infinite? > > What does it mean for a single particular value to be "potentially" > whatever? > "A particular value is potentially whatever" is a contradiction in terms. I believe you know what potential infinity is, and playing as if I'm stupid, or you are, won't work. It's what WM believes in, as opposed to actual infinity. It's what you call "countably infinite", even though it's finite. Play smart. You don't have to be nice, but don't be dumb. So, answer the question you promised to, or show how I didn't answer yours. Touche! TO
From: mueckenh on 19 Aug 2006 16:27 David R Tribble schrieb: > mueckenh wrote: > > Learn: Infinitely many stairs require infinite height. But that it not > > admitted in mathematics. There have to be infinitely many stairs which > > give an infinitely long staircase ... > > Yes, infinitely many finite-sized steps produce a staircase of infinite > height. That is admitted in mathematics. > > > ... but this staircase is allegedly of finite height. > > No, the staircase is infinite in height, but each step is finite in > height. The distance between any two steps is finite, but the total > height of all the steps together (the entire staircase) is infinite. The staircase is not an independent individual, but it is nothing than the set of stairs. Without a stair surpassing the height H the staircase cannot surpass height H. Without a stair surpassing every finite height the stair cannot surpass every finite height. (surpassing every finite = beng infinite). But the length surpasses every finite length. Regards, WM
From: mueckenh on 19 Aug 2006 16:29
imaginatorium(a)despammed.com schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > > > > > > > >> > Therefore there are not infinitely many difference[s] of 1 > > > >> > between natural numbers. > > > >> > > > >> Your consequent is proven false (see below). Therefore your > > > >> implication is false, too. > > > > > > > > You are in error. > > > > > > Where precisely is the error? > > > > The assertion that infinitely many differences of 1 can be provided by > > finite natural numbers. Why? Here it is: The staircase is not an independent individual, but it is nothing than the set of stairs. Without a stair surpassing the height H the staircase cannot surpass height H. Without a stair surpassing every finite height the staircase cannot surpass every finite height. (surpassing every finite = infinite). But the length surpasses every finite length. > > You've been repeating this moronic nonsense for so long it's obviously > hopeless, but just answer me a simple question. In your view of these > "finite natural numbers", I presume 1 is included, and 2, and 3, and so > on. If you were to simply count them, 1, 2, 3, 4, and so on, never > stopping unless you came to an end, do you in fact think you _would_ > reach an end? Any suggestions as to what this end would look like, > given that it would entail a natural number such that adding 1 somehow > failed to happen. No. This moronic nonsense would only be necessary if the complete set of natural numbers would actually exist and have a cardinal number larger than any natural number. Regards, WM |