Cantor Confusion
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From: Virgil on 4 Dec 2006 14:56 In article <1165239659.765379.43960(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Yes. But the mapping is not the usual one (one edge --> one path). The > > > edges are subdivided in shares. > > > > Then it does not establish a one-to-one relationship. In one-to-one you > > must match whole with whole and not fractionate. > > Yes, so people think, franticly closing their eyes in front of a > fractional relation. Bijections are one-to-one matchings, not fraction to fraction matchings. You must demonstrate whole to whole if you are to demonstrate them at all. > > But in order to kill your prejudice: Map the edges on the paths one-to > one by random choice. > > > 0. > /a \ > 0 1 > /b \c / \ > 0 1 0 1 > ............................ > > Map edge a on one of the paths beginning with 0.0 (for instance on > 1/3). > Map edge b on one of the paths beginning with 0.00, but not on that one > which carries edge a. > Map edge c on one of the paths beginning with 0.01, but not on that one > which carries edge a. > > By this random mapping, we have a bijection from the set of edges onto > the set of path. This is proved by the fact that always, when two paths > become separated, one edge is available to be mapped on a path not yet > carrying an edge. No one objects to any number of INjections of the set of edges into the set of paths, which is what WM outlines above, but injections are not all surjections, and this injection is not a surjection.
From: Virgil on 4 Dec 2006 14:59 In article <1165240030.468931.322160(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Therefore we can denote a set by X and we can say that the set X grows. > > > > When one calls X a variable, that does not mean that its value varies > > but that its value can be any member of some set of allowable values. > > That is one of several interpretations. In this interpretation there is > no place for potential infinity. This is the interpretation used in mathematics. To the extent that one rejects this interpretation, one is rejecting mathematics. >That is a good reason to reject it. Then WM is declaring that he does not want anything to do with actual mathematics.
From: Eckard Blumschein on 4 Dec 2006 15:04 On 12/1/2006 12:53 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> >> >> Likewise oo + a = oo seems to imply a = oo - oo = 0. > > oo is not a number. Correct. There are people who extend the reals to include oo. This is consequent because real numbers are strictly speaking also no numbers. > > Once again you show you simply do not understand mathematics. > > Bob Kolker Let's exchange arguments rather than insults.
From: Eckard Blumschein on 4 Dec 2006 15:09 On 12/1/2006 12:51 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> Once again: >> _Sentences_ _containing_ _unlimited_ _quantifiers_ _are_ _in_ _general_ >> _meaningless_" > > Nonesense! Unsinn! Quoted from the mentionend book. I see that the formulation is mistakable, maybe deliberately chosen by the authors. The essence is however correct. > > (x)(T(x) -> P(x)) > > T(x) means x is a tree, P(x) means x is a plant and -> is implies.
From: MoeBlee on 4 Dec 2006 15:23
Eckard Blumschein wrote: > Correct. There are people who extend the reals to include oo. Would you give an example of a text that does this? What we sometimes do is add two points (called 'oo' and '-oo') to the real number system so that we have a different, extended system (which is not a complete ordered field). But that does not meant that we consider oo and -oo to be real numbers. MoeBlee |