Cantor Confusion
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From: mueckenh on 23 Dec 2006 16:27 Newberry schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > You need to construct a surjection. > > > > No, I don't. > > > > > > Yes you do, if you want to show that there exists a surjection > > > from edges to paths. > > > > You cannot construct a surjection because irrational numbers do not > > exist. > > Are you saying that x^2 = 2 does not have a solution? Yes. But that is another topic. (In short: There are less than 10^100 bits in the universe. Therefore, no irrational number can be approximated better than to about 10^-100. The Cauchy criterion for converging series fails.) Regards, WM
From: Ralf Bader on 23 Dec 2006 16:53 mueckenh(a)rz.fh-augsburg.de wrote: > > Newberry schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > > > You need to construct a surjection. >> > > > No, I don't. >> > > >> > > Yes you do, if you want to show that there exists a surjection >> > > from edges to paths. >> > >> > You cannot construct a surjection because irrational numbers do not >> > exist. >> >> Are you saying that x^2 = 2 does not have a solution? > > Yes. But that is another topic. (In short: There are less than 10^100 > bits in the universe. Therefore, no irrational number can be > approximated better than to about 10^-100. The Cauchy criterion for > converging series fails.) No, it does not fail. You could have seen this criterion applied in any introductory analysis text, if you ever had bothered to look into one before gossiping stupid nonsense.
From: Newberry on 23 Dec 2006 16:57 mueckenh(a)rz.fh-augsburg.de wrote: > Newberry schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > You need to construct a surjection. > > > > > No, I don't. > > > > > > > > Yes you do, if you want to show that there exists a surjection > > > > from edges to paths. > > > > > > You cannot construct a surjection because irrational numbers do not > > > exist. > > > > Are you saying that x^2 = 2 does not have a solution? > > Yes. But that is another topic. (In short: There are less than 10^100 > bits in the universe. Therefore, no irrational number can be > approximated better than to about 10^-100. The Cauchy criterion for > converging series fails.) What cannot be approximated? > > Regards, WM
From: Dik T. Winter on 23 Dec 2006 18:52 In article <1166845904.426550.122020(a)48g2000cwx.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > stephen(a)nomail.com wrote: .... > > > What is wrong with WM's mapping? You divide each edge into two halves > > > and pass one half to each branch. Then you divide the passed half again > > > and again. > > > > So which edge is mapped to the path that always goes left? > > Remember, you need to uniquely map each path to an edge. > > No, I don't. What I am showing here is that each path will accumulate > the weight of two edges as it approaches infinity. No, you do not. In the finite case you do, approximately. In the infinite case you do not because *all* parts you pass to a path are reduced to size 0. Consider what we pass on to the paths that go left only. Look at the following: level 1: 1 edge at level 1 level 2: 1/2 edge at level 1 + 1 edge at level 2 level 3: 1/4 edge at level 1+ 1/2 edge at level 2 + 1 edge at level 3 etc. for each finite n, but what about the infinite case? level 'oo': 1/oo edge at level 1 + 1/oo edge at level 2 + ... Where do you see the 2 coming up? It is certainly *not* the series 1 + 1/2 + 1/4 + ..., because there is *no* edge that is passed in full, there is also *no* edge of which 1/2 is passed, etc. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 23 Dec 2006 18:59
In article <1166908153.383999.155270(a)80g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > I stated that there is no possible way to construct a well-ordering. > > > You opposed. You cannot sustain your statement. That's all. > > > > Given AC, there is an Hamel basis and using that a well-ordering can > > be defined. > > No Dik, it has been proved that a definable well-order is impossible to > accomplish. It is impossible if you do not assume AC. Where has it been proven that it is impossible if you *do* assume AC? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |