An uncountable countable set
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From: Virgil on 15 Aug 2006 15:42 In article <1155664145.962603.169560(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155487078.836213.291820(a)p79g2000cwp.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1155314675.845888.174190(a)i3g2000cwc.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > Virgil schrieb: > > > > > > > > > > So we have infinitely many finite triangles, but without that > > > > > > "final > > > > > > edge", no infinite triangle. > > > > > > > > > > Cantor's list has no final line. Is it finite? > > > > > > > > Since a triangle requires 3 edges, without all 3 it is not a triangle. > > > > An endless list does not require an end, so is "complete" without one. > > > > > > A symmetric rectangular triangle is completely determined by one edge > > > next to the right angle. > > > > But edges, as sides of triangles, being segments, must have two ends, > > and that "edge" does not. > > The edge has a definite size, it is a well defined quantity (according > to Cantor). Edges in geometry must have endpoints and triangles must have vertices. "Mueckenh"'s "triangle" has only one vertex, which makes it not a triangle.
From: Virgil on 15 Aug 2006 15:46 In article <1155664307.522509.91390(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Of course. There is no actual infinity. There are more things in heaven and earth, "Mueckenh", than are dreamt of in your philosophy.
From: Virgil on 15 Aug 2006 15:56 In article <1155664424.216852.321590(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Be sure, even the natural numbers cannot be enumerated, because a > natural number can be added to every list of natural numbers. My definition of "enumerating a set" is satisfied by creation of a surjective function from N to to that set (or an injective function from that set to N). The function f:N -> N: n |-> n enumerates N. If E is the set of even naturals numbers then g:N -> E : x |-> 2*x ennumerates E.
From: Virgil on 15 Aug 2006 15:58 In article <1155664465.391460.103870(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > An infinite sum of 1's is not infinite? > > > > n > > lim sum 1 = lim n =def L > > n -> oo i = 1 n -> oo > > > > There is no such L in N. > > Correct. Therefore there are not infinitely many difference of 1 > between natural numbers. Non sequitur. The non convergence of a sequence does not disprove the existence of the sequence.
From: mike4ty4 on 15 Aug 2006 16:04
"mueck...(a)rz.fh-augsburg.de " wrote: > An uncountable countable set > > There is no bijective mapping f : |N --> M, > where M contains the set of all finite subsets of |N > and, in addition, the set K = {k e |N : k /e f(k)} of all natural > numbers k which are mapped on subsets not containing k. > > > This shows M to be uncountable. > > > Regards, WM A set can't be both uncountable and countable at the same time just like a statement can't be both true and false at the same time. Case closed right there. Ha! |