Cantor Confusion
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From: mueckenh on 14 Oct 2006 16:41 Virgil schrieb: > >This statement does not refer in any respect to > > the number of digit positions. All finite digit positions are covered > > by one finite number M. > > What about position M + 1? > All finite digit positions N =< M+1 are covered by the finite unary number M+1. You will fail in your attempt to find a finite number up to which all numbers cannot be covered by one single unary number. That should show you that your opposite assertion is wrong. We have: For finite unary numbers in a linear problem the quantifiers can be interchanged. > > > > > > > No, the fact that every digit position of 0.111... is finite does not > > > mean that M does not depend on N, and that is what we need. > > > > Why should we need that? > > Because for every M there is an N = M+1 which M does not cover. > But it is finite, no matter how you name it. > > > > The problem is that you believe in an actually infinite series of > > finite numbers. This belief leads to intermingling digit position and > > number of digits just according to the needs and, after all, it leads > > to such self-contradictive statements as we have seen here with "the > > vase". > > There is nothing self contradictory in "the vase" but there are > contradictions between the axiom systems on which "the vase" is based > and the assumptions of those who oppose those axiom systems. > > According to the ZFC system: The vase is empty at noon, because all natural numbers left it before noon. By means of the ZFC system we can formulate sequences and their limits in mathematical language. From this it follows that lim {n-->oo} n > 1. And from this it follows that the vase is not empty at noon. You will and must disagree, but the spectators will at least get to know this inconsistency. Regards, WM
From: Virgil on 14 Oct 2006 16:41 In article <1160814654.862306.286450(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > 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 > > University of Applied Sciences, Augsburg. My sympathies to his poor students. > > - 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". > > Is the Imaginator too simple minded to understand, or is it just an > insult? The quantifier interchange is impossible in general, but it is > possile for special *linear* sets in case of *finite* elements. For example? Does "Mueckenh" claim that, say, "For every natural n there is a natural m such that m > n" and "There is a natural m, such that for every natural n, m > n" are logically equivalent? All the elements are finite and linearly ordered.
From: mueckenh on 14 Oct 2006 16:48 Alan Morgan schrieb: > >But sqrt(-1) does not yield contradictions, as far as I know. > > It violates the heck out of my intuition. The objections to set > theory seem to arise from someone's dislike for the conclusions > or an inability to do mathematics correctly. Is Weyl this someone? "...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." > No one has actually > shown a contradiction yet. To be repeated 10 times per day, in order to fix the belief of those adherents which have critical minds. (It is the rumor that such ones do in fact exist, but nobody knows where.) Regards, WM.
From: Virgil on 14 Oct 2006 16:52 In article <1160814799.951385.291960(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil 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. > > > > 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 > > You misunderstood the notation (even after so many postings!). There is > only one edge a taken as an example. The edge on the right hand side is > not labelled. Every edge must have its own label if they are to be referred to by label. How about L and R a labels for the left and right branches at the root node, LL and LR for the left and right branches at the left node and RL RR for the rightmost pair at the right node , then LLL, LLR; LRL, LRR; RLL, RLR; RRL, RRR for edges at the next level, and so on ad infinitum. That gives every edge in the entire tree a unique label by which it may be referenced.
From: Virgil on 14 Oct 2006 16:56
In article <1160814979.041963.233270(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160652854.196211.117740(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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. > > You stated so, but your statement is false. Not in my world. > > 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. > > Why "no further"? There is no reason to insert that. That you do not see a reason does not mean that there isn't one. A set of unary numbers which covers 0.111... to every finite position cannot be replaced by a single finite unary number. |