An uncountable countable set
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From: Virgil on 18 Aug 2006 22:30 In article <ec5jt1$4gr$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > 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? Trust Tony to draw the one wrong conclusion.
From: Virgil on 18 Aug 2006 22:31 In article <ec5k1i$4gr$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > By the way, hello! Nice to see you all again, I think. Does Tony actually think?
From: imaginatorium on 19 Aug 2006 06:35 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! I'll assume this is a question: Q1: "Is the greatest possible value that any natural number can reach infinite in height?" I'll tell you the answer to Q1 when you tell me the answer to Q2... 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. Q2: "Is the sharpest corner of a circle less than or greater than 90 degrees?" 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? Isn't The width the same as the > height of the square, as drawn? I say the set, as defined, is the same > size as its maximum element. > > :)
From: mueckenh on 19 Aug 2006 10:41 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. > > > You just proved it to be true. > > You may "just" as well prove the opposite. Please do so. > > > The set of natural numbers (i.e., finite numbers n, i.e., numbers > > with finitely many differences of 1 between 1 and n) does not yield > > infinitely many differences of 1. > > This is a reiteration not a proof. There are not infinitely many differences, if we consider the finity of each number. There are infinitely many differences, if we consider infinitely many numbers. Therefore one of the assumptions is wrong: Either there are not infinitely many numbers or there is at least one number which is infinite by size. > > Do you agree that P ~ omega? P has the same cardinality as |N. But either there are not infinitely many numbers or there is at least one number which is infinite by size. > Do you agree that card (omega) /e omega? card (omega) = aleph_0 = the number of natural numbers and that cannot be infinite, as we have seen. Regards, WM
From: mueckenh on 19 Aug 2006 10:42
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 refuse to understand what "to exist" means. In no case infinitely many differeces of 1 can exist unless infinitely many diferences of 1 do exist. But that means an infinite size. Regards, WM. |