An uncountable countable set
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From: Virgil on 15 Aug 2006 15:32 In article <1155487181.028039.253190(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155314605.714007.167990(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > The line number is infinite by the axiom of infinity. > > > > Where in the axiom of infinity are infinite number lines mentioned? > > > > > If no line is numbered as omega, where can we find the line omega + 1? > > > > Who says there is one? > > Set theory. > > > > > > > > Thus "mueckenh" is claiming to be able to inject the power set of the > > > > naturals into the set of naturals. > > > > > > > > We should be interested in seeing his attempts to perform this > > > > impossibility. > > > > > > Where does *my arguing* fail? > > > 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. > > > > >The set of paths is uncountable, but as we have seen, it > > > contains less elements than the set of edges. > > > > " What is > > true only for finite cases need not be true for infinite ones. > > What is true up to every level n of the tree is true for the whole > tree. Only if the tree is finite. > > Else: What is true for a finite segment of Cantor's list need not be > true for the whole infinite list. Quite so. What is true for every finite segment is that it is a finite segment. That is not true of the whole infinite list. Thus that ARE things true for every finite sublist that are not true for the entire infinite list. similarly for trees.
From: Virgil on 15 Aug 2006 15:35 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 one does not have either such edge. Neither the "bottom" edge nor the "left edge" exist as an edge, as neither can have two endpoints to the hypothetical "edge".
From: Virgil on 15 Aug 2006 15:38 In article <1155068921.849102.10570(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1154967573.942270.13030(a)p79g2000cwp.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Example: > > > The third 1 of 0.111... can index the third 1 of 3 = 0.111 as well as > > > the third one of 5 = 0.11111. > > > > > > Since the third 1 and the fourth 1 are identical, absent their > > predecessors, neither can index anything by itself. > > > > If "mueckenh" wants the string of the first three 1's as index for the > > third 1, that is possible. > > > > > > > > > > > > If 1/9 is not in the list > > > > > but > > > > > can be indexed completely by list numbers > > > > Each digit in the decimal expansion can be indexed by a natural number > > but the 'completed' expansion can only be "indexed" by the 'completed' > > set of all naturals. Which is perfectly symmetric. > > > > > > , this symmetry is broken, > > > > > because: What means "not in the list"? It means that 1/9 has more > > > > > 1's > > > > > than every list number. > > > > It does not mean that 1/9 has a decimal expansion with "more" 1's than > > there are naturals in the set of all naturals. > > All numbers with as many 1's as are indexible by naturals are in the > true list. 0.111... is not in the true list. That is proof enough. Since one can index 0.111... first and shift the indexing of each digit to the next digit, all are indexed including 0.111....
From: Virgil on 15 Aug 2006 15:39 In article <1155663962.465480.57580(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > An infinite sum of 1's is not infinite? > > > > "Mueckenh" again proves that he cannot read. It is false and stupid to > > say that an infinite sum being infinite prohibits infiniteness. > > Don't misquote me. It prohibits infiniteness of the set of natural > numbers. > > > But > > saying that in no way implies that an infinite sum of 1's is not > > infinite. > > But it is not a natural number. So? There are a lot of things that are not natural numbers. So what? > > Regards, WM
From: Virgil on 15 Aug 2006 15:40
In article <1155664057.385235.133360(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155485742.529974.141050(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Of course you can index each n. But your "each n" stems from the true > > > list. And we know that 0.111... is not in the true list, because it is > > > distinguished from any element of the true list. > > > > It can be indexed by the first 1, and each 1 indexed by the next 1 and > > everything is then indexed. > > Everything which is in the true list > > 0.1 > 0.11 > 0.111 > ... > > can be indexed. But nothing which is not in the true list. maybe not by you, but Hilbert, and others, can. Since one can index 0.111... first and shift the indexing of each digit to the next digit, all are indexed including 0.111.... |