Cantor Confusion
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From: mueckenh on 9 Oct 2006 11:19 Dik T. Winter schrieb: > In article <1160404669.240794.298920(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > The balls in vase problem suffers because the problem is not well-defined. > > > Most people in the discussion assume some implicit definitions, well that > > > does not work as other people assume other definitions. How do you > > > *define* the number of balls at noon? You can not use limits, because the > > > limit does not exist when you use standard mathematics. > > > > But we can safely say that lim{n-->oo}n = 0 is false. > > Yes, it is false because that liit does not exis. > > > lim{n-->oo}n can be estimated by lim{n-->oo} 1/n = 0. > > You can not estimate something that does not exist. > > > So using standard > > > definitions there is no answer. More precise, given the sequence of sets: > > > {1, ..., 10) > > > {2, ..., 20} > > > {3, ..., 30} > > > etc., is there a limit? Well, no, there is no defined limit unless you > > > define what a limit of sets looks like. I have never seen a definition > > > that tells me how the limit of a sequence is defined. The limit of the > > > size of the sets also gives no answer, because that limit does not exist. > > > Strange enough, when somebody goes on to define things, *you* question his > > > definitions, rather than the result. > > > > The limit {1,...,n} for n-->oo is N, if N does exist. > > By what definitions? By the definition of the limit ordinal omega (= N). Regards, WM
From: David Marcus on 9 Oct 2006 12:23 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David Marcus schrieb: > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Hi, Dik, > > > > > > > > > > > > > > I would like to publish our result to the mathematicians of this group > > > > > > > in order to show what they really are believing if they believe in set > > > > > > > theory. > > > > > > > > > > > > > > There is an infinite sequence S of units, denoted by S = III... > > > > > > > > > > > > > > This sequence is covered up to any position n (included) by the finite > > > > > > > sequences > > > > > > > I > > > > > > > II > > > > > > > III > > > > > > > ... > > > > > > > > > > > > What do you mean by "cover"? > > > > > > > > > > A covers B if A has at least as many bars as B. A and B are unary > > > > > representations of numbers. > > > > > > > > > > Example: A = III covers I and II and III but not IIII. > > > > > > > > > > > But it is impossible to cover every position of S. > > > > > > > > > > > So: S is covered up to every position, but it is not possible to cover > > > > > > > every position. > > > > > > > > So, your conclusion is that no finite sequence of I's will cover S. > > > > Correct? > > > > > > > > Is this your entire theorem or is there more to the conclusion? > > > > > > My conclusion is: > > > Either > > > (S is covered up to every position <==> S is completely covered by at > > > least one element of the infinite set of finite unary numbers <==> S is > > > an unary natural) ==> Contradiction, because S can be shown to be not a > > > unary natural. > > > > Are you saying that standard mathematics contains a contradiction > > Yes, obviously. You wrote that "A covers B" means that A has at least as many bars as B. Does "S is completely covered by at least one element of the infinite set of finite unary numbers" mean that S is covered by an A that has a finite number of bars? -- David Marcus
From: David Marcus on 9 Oct 2006 12:31 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > > > empty. > > > > > 2) Before and at noon there are more balls in the vase than have come > > > > > out. > > > > > > > > How do you translate the words of the problem into mathematics? > > > > > > 0) There is a bijection between the set of balls entering the vase and > > > |N. > > > 1) There is a bijection between the set of escaped balls and |N. > > > 2) There is a bijection between (the cardinal numbers of the sets of > > > balls remaining in the vase after an escape)/9 and |N. > > > > > > Instead of "balls", use "elements of X where X is a variable". > > > > Sorry, perhaps I wasn't clear. That's not what I meant. Please state the > > problem using mathematics. > > I am sure you are able to translate brief notions like "to enter, to > escape" etc. by yourself into terms of increasing or decreasing values > of variables of sets, if this seems necessary to you. Here, without > being in possession of suitable symbols, it would become a bit tedious. Yes, I can translate it myself. However, that would only tell me how I interpret the problem. Until you tell me how you would translate it, I don't know how you are interpreting the problem. Before we can draw any mathematical conclusions, we need to know what mathematical problem we are discussing. If you prefer, I could offer a translation and you could tell me if it is what you mean. -- David Marcus
From: Virgil on 9 Oct 2006 12:52 In article <b7f51$452a1029$82a1e228$25909(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Mathematicians have found another name for scientitic facts. They call > them "just an opinion". Mathematicians do not contest the alleged factualness of scientific "facts", but do contest their relevance in determining what mathematicians are to be allowed to think. > No! The burden is yours. _You_ have to provide > arguments why it is admissible to allow for infinite sets. Because we can! > While _all_ > eyes and all instrumentation in the cosmos can only make observations > of things that are _finite_. That presumes something hypothesized but not yet established, that the cosmos is itself finite.
From: Virgil on 9 Oct 2006 12:53
In article <723a8$452a1312$82a1e228$25909(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > A.S.S. needs a more interesting axiom set to interest me. > > You don't get the message. Do you? > > Han de Bruijn Not yours, at all events. |