Cantor Confusion
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From: imaginatorium on 12 Oct 2006 15:09 David Marcus wrote: > Dik T. Winter wrote: > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > To inform the set theorist about the possible existence of sets with > > > finite cardinality but without a largest number. > > > > Interesting but in contradiction with the definition of the concept of > > "finite set". So you are talking about something else than "finite > > sets". > > It would seem he is. I don't understand why people use words in non- > standard ways without explaining what they mean. They are guaranteeing > that no one will understand them. Several possible obvious answers. (BTW, it was fairly clear you were new around here -- then you tried asking Ross Finlayson what he means. Clinched it.) (a) The writer is playing a bizarre game of trollery. (b) The writer is simply misinformed about the meaning of a particular term. (Don't think this is common) (c) The writer does not have the mental apparatus to understand a formal argument, and therefore simply cannot comprehend the difference between a number of statements of subtle difference. This seems to be most common. For example, Mueckenheim - who astonishingly appears to *teach* mathematics at some sort of college in Germany - plainly cannot comprehend the difference that swapping quantifiers makes. He cannot comprehend that there might be a difference between the significance of "every" in "Every girl in the village has a lover" and "John makes love to every girl in the village". Good luck (not that it will get you anywhere). Brian Chandler http://imaginatorium.org
From: Virgil on 12 Oct 2006 15:35 In article <1160651933.936855.275470(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Please state an internal contradiction of set theory. Please use the > > standard language of set theory/mathematics so that we can understand > > what the contradiction is without needing to ask what all the words > > mean. > > Good heavens, there are so many. Where shall I start with? All of them involve assumptions ma"Mueckenh"kes that are not any part of any standard axiom set. > > 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. One can relate one of the 'a' edges, say the left one, to all paths beginning with 0.0 and the other 'a' edge, the right one with. the string beginning 0.1 Then the left b edge is 0.00, the right b edge is 0.01, the left c edge is 0.10, and the right c edge is 0.11 And at each level a left branching edge appends a 0 and a right branching edge appends a 1 to the string Then every path truncated after a finite number of steps gives an edge belonging to that path. >Continuing in > this manner in infinity, we see that every single infinite path is > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, I do not see this at all, as "Mueckenh"'s method is not telling us which edge a path takes at any node, so does not identify paths at all. A path consists of an infinite sequence of branchings, either left edge or right edge at each node, and not merely fractional parts of edges. "Mueckenh" cannot show his "model" of a binary tree by any construction strictly within ZFC or NBG.
From: Virgil on 12 Oct 2006 15:44 In article <463c0$452e265e$82a1e228$30902(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > mueckenh(a)rz.fh-augsburg.de wrote in response to Virgil: > > > For the vase problem with the number n(t) of balls in the vase after t > > transactions we can find always a positive eps such that for t > t_0: > > 1/n(t) < eps, hence n(t) larger than an arbitrary positive number. > > > > Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. > > Precisely! Virgil doesn't know how to handle limits. > > Han de Bruijn I merely note that there is no requirement in the problem that the limit be the value at noon. We have an increasing sequence of times t_n which converge to noon so lim_{n in N} t_n = noon We also have a number of balls b(t_n) = 9*n You are claiming , essentially that all limits exist and lim_{n in N} b( t_n ) = b( lim_{n in N} t_n ) That would require that b(t) be continuous at t = noon, but nothing in the problem either requires nor even specifically allows this.
From: Virgil on 12 Oct 2006 15:48 In article <1160652854.196211.117740(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > > 0.1 > > > > > > 0.11 > > > > > > 0.111 > > > > > > ... > > > > > > > > > > > > > > That is correct. But every element of the natural numbers is finite. > > > > > Hence every element covers its predecessors. If 0.111... is covered by > > > > > "the whole list", then it is covered by one element. That, however, is > > > > > excuded. > > > > > > > > > > > > > Since no one has claimed that '0.111... is covered by "the whole > > > > list"', I fail > > > > to see the relevence of a sentence that starts out > > > > 'If 0.111... is covered by "the whole list"'. > > > > > > If every digit position is well defined, then 0.111... is covered "up > > > to every position" by the list numbers, which are simply the natural > > > indizes. I claim that covering "up to every" implies covering "every". > > > > > Quantifier dyslexia. > > Quantifier magic may apply and may be useful at several occasions. But > to state that in a unary representation of natural numbers the union of > "up to every" and "every" have different meaning is easily disproved. > > > The fact that > > > > for every digit position N, there exists a natural number, M, > > such that M covers 0.111... to position N > > > > does not imply > > > > there exist a natural number M such that for every digit > > position N, M covers 0.111... to position N > > In case of linear sets we have a third statement which is true without > doubt: > > 3) Every set of unary numbers which covers 0.111... to a finite > position N can be replaced by a single unary number. As stated it is false. Try this: 3') Every set of unary numbers which covers 0.111... to a finite position, n, and no further can be replaced by a single unary number. > > This holds for every finite position N. If 0.111... has only finite > positions, then (3) holds for every position. As 0.111... does not > consist of mre than every position, it holds for the whole number > 0.111.... False!
From: Virgil on 12 Oct 2006 15:52
In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Completed infinity > does not exist. There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy. |