Cantor Confusion
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From: Virgil on 16 Oct 2006 17:23 In article <1161029578.356913.74630(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > No. I will not have a largest number unless I have chosen *how* to use > > > all the bits. In unary representation the largest number will clearly > > > be less than 10^100. But I am not indebted to focus on any fixed > > > representation. Therefore, there is no largest number. > > > > Then there is no set of numbers. > > Weird is that adding 9 balls instead of 1 per transaction leads to zero > balls. Even weirder is that removing every ball leaves some. > > Use f(t) = 1/9t for t e N, the t-th transaction. Use t for time as time increases towards noon. For every ball, n, there is a time of insertion before noon, ti_n and a time of removal before noon, tr_n, with the time of removal never before the time of insertion and always before noon. I.e., ti_n <= tr_n < noon. In addition ti_n <= ti_{n+1} for all n and tr_n < tr_{n+1} for all n. Does anyone dispute that all this is valid for the original gedankenexperiment? Unless someone can dispute it, how can anyone claim that there are any balls in the vase at noon?
From: Virgil on 16 Oct 2006 17:24 In article <1161029898.290793.7290(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > I gave my translation of the ball and vase problem. I don't see any > > contradictions that follow from it. If you say it implies two > > contradictory conclusions, please state them and your proof. > > What was your result? How many balls are in the vase at noon according > to your translation? For every ball, n, there is a time of insertion before noon, ti_n and a time of removal before noon, tr_n, with the time of removal never before the time of insertion and always before noon. I.e., ti_n <= tr_n < noon. In addition ti_n <= ti_{n+1} for all n and tr_n < tr_{n+1} for all n. Does anyone dispute that all this is valid for the original gedankenexperiment? Unless someone can dispute it, how can anyone claim that there are any balls in the vase at noon?
From: MoeBlee on 16 Oct 2006 17:27 Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee schreef: > > > Han de Bruijn wrote: > > > Sigh! Start digging into my website. I've said more about mathematics > > > than anybody else in 'sci.math'. > > > > That's hilarious! I didn't even have dig at all to find you proposing > > an inconsistent set of axioms and blaming not yourself but set theory > > for the inconsistency - on the very first page I saw at that web site! > > Dig deeper! "Dig deeper." It has the sound of a punchline to a joke.
From: MoeBlee on 16 Oct 2006 17:31 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > > Han de Bruijn wrote: > > > > > > How can we know, heh? Can things in the real world be true AND false > > > (: definition of inconsistency) at the same time? > > > > That is not the definition of "inconsistency" in Mathematics. On the > > other hand, I don't know of any statements in Mathematics that are both > > true and false. If you have one, please state it. > > What then is the precise definition of "inconsistency" in Mathematics? How many times does it have to be posted? G is inconsistent <-> G is a set of formulas such that there exists a formula P such that P and its negation are both members of G. MoeBlee
From: Virgil on 16 Oct 2006 17:32
In article <1161030262.984508.214180(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > > > > Consider the binary tree which has (no finite paths but only) > > > > > > > infinite > > > > > > > paths representing the real numbers between 0 and 1. The edges > > > > > > > (like a, > > > > > > > b, and c below) connect the nodes, i.e., the binary digits. The > > > > > > > set of > > > > > > > edges is countable, because we can enumerate them > > > > > > > > > > > > > > 0. > > > > > > > /a \ > > > > > > > 0 1 > > > > > > > /b \c / \ > > > > > > > 0 1 0 1 > > > > > > > ..................... > > > > > > > > > > > > > > Now we set up a relation between paths and edges. Relate edge a > > > > > > > to all > > > > > > > paths which begin with 0.0. Relate edge b to all paths which > > > > > > > begin with > > > > > > > 0.00 and relate edge c to all paths which begin with 0.01. Half > > > > > > > of edge > > > > > > > a is inherited by all paths which begin with 0.00, the other half > > > > > > > of > > > > > > > edge a is inherited by all paths which begin with 0.01. > > > > > > > Continuing in > > > > > > > this manner in infinity, we see that every single infinite path > > > > > > > is > > > > > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related > > > > > > > to any > > > > > > > other path. > > > > > > > > > > > > Are you using "relation" in its mathematical sense? > > > > > > > > > > Of course. But instead of whole elements, I consider fractions. That > > > > > is > > > > > new but neither undefined nor wrong. > > > > > > > > > > > > Please define your terms "half an edge" and "inherited". > > > > > > > > > > I can't believe that you are unable to understand what "half" or > > > > > "inherited" means. > > > > > I rather believe you don't want to understand it. Therefore an > > > > > explanation will not help much. > > > > > > > > In standard terminology, a "relation between paths and edges" means a > > > > set of ordered pairs where the first element of a pair is a path and > > > > the > > > > second is an edge. Is this what you meant? > > > > > > Yes, but this notion is developed to include fractions of edges. > > > > > > > > I am at a loss as to how "half an edge" can be "inherited" by a path. > > > > > > An edge is related to a set of path. If the paths, belonging to this > > > set, split in two different subsets, then the edge related to the > > > complete set is divided and half of that edge is related to each of the > > > two subsets. If it were important, which parts of the edges were > > > related, then we could denote this by "edge a splits into a_1 and a_2". > > > But because it is completely irrelevant which part of an edge is > > > related to which subset, we need not denote the fractions of the edges. > > > > > > > As > > > > far as I know, a path is a sequence of edges. Is this what you mean by > > > > "path"? > > > > > > Yes. A path is a sequence of nodes (bits, 0 or 1). The nodes are > > > connected by edges. > > > > > > > > Is edge a the line connecting 0 in the first row to 0 in the second > > > > row? > > > > > > Correct. This edge is related to all paths starting with 0.0. So half > > > of edge a is related to all paths starting with 0.00, and half of it is > > > related to all paths starting with 0.01 > > > > > > > Or, is it the line connecting 0 in the first row to 1 in the second > > > > row? > > > > Or, is it something else? > > > > > > I have tried above to make it a bit clearer. > > > > First you refer to a "relation between paths and edges". > > Correct. > > > Then you say > > the "edge is related to a set of path". > > Is there any contradiction? > > > Then you say that "half of that > > edge is related to each of the two subsets". I'm sorry, but I can't > > follow this. > > I warned you that this point is new: The edges are split in shares of > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > most people may have had the same problems as you today. I am sure you > can understand it from the written text above. (Many others have > already understood it.) No one has explicitly admitted understanding it, it is only that not everyone has claimed not to. I do not understand how one can "split" edges in a supposed bijection between edges and paths. In the sort of bijections I am used to, the objects being bijected with each other must exist as unsplit wholes with one whole unsplit object being matched with another whole unsplit object until all the whole unsplit objects of each set have been paired off. Where does "Mueckenh" find any definition of bijection that allows parts of elements to be matched up with anything? |