Cantor Confusion
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From: mueckenh on 19 Dec 2006 07:09 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > [...] > >> Assume that there exists an L_D which contains all the numbers > >> contained in the diagonal. L_D is bounded, > > > > If an unbounded diagonal exists, then obviously an unbounded line must > > exist. > > The conclusion is false, so the antecedent cannot be true. > > Non sequitur: > > Logical Implication > p q p -> q > F F T *1) > F T T > T F F *2) > T T T > I did not prove the truth of the implication (which is true from the definition of the diagonal in the EIT and from the linear ordering of the finite elements of N). I proved that the premise is false. Take a bit time to understand it. If not, take a bit longer. If not at all, try to study something other than math. Regards, WM
From: William Hughes on 19 Dec 2006 07:53 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > Albrecht wrote: > > > > but in the same time |N isn't complete > > > since |N is bijectable to the diagonal and the diagonal is never > > > complete since there is no complete line (number) which covers the > > > diagonal. > > > > No, the claim is that there is no line that covers the diagonal. > > Which results in the conclusion that there is no diagonal. No. Only properties of the diagonal that you have accepted are used. The conclusion is that there is no line. - William Hughes
From: Franziska Neugebauer on 19 Dec 2006 07:54 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > William Hughes schrieb: >> [...] >> >> Assume that there exists an L_D which contains all the numbers >> >> contained in the diagonal. L_D is bounded, >> > >> > If an unbounded diagonal exists, then obviously an unbounded line >> > must exist. [(*)] >> > The conclusion is false, so the antecedent cannot be true. >> >> Non sequitur: >> >> Logical Implication >> p q p -> q >> F F T *1) >> F T T >> T F F *2) >> T T T >> > I did not prove the truth of the implication 1. Yes. You did _not_ consistently derive the implication [(*)]. Alas for modus tollens the validity of the implication is mandatorily required. > (which is true from the definition of the diagonal in the EIT and from > the linear ordering of the finite elements of N). 2. Neither a _definition_ of "diagonal" nor a definition of "linear ordering" constitute a proof (consistent derivation) of (*) from the suppositions. 3. What does "which" in "which is true from the definition ..." refer to? > I proved that the premise is false. 4. You did not. Modus tollens requires p->q to be valid. Instead you simply composed an implication consisting of two unrelated assertions of which the latter, q is false, which you seem to agree to. ,----[ <458702c7$0$97240$892e7fe2(a)authen.yellow.readfreenews.net> ] | *2) is the interpretation I would prefer. I may help you: | | If the moon exists today then obviously Germany must be | a monarchy today. | | Following the M�ckenheim-reasoning one would "conclude" that the moon | does not exist. The point is that you erroneously think by simply | juxtaposing p, "->" and q you "prove truth" on the implication "p->q". `---- > Take a bit time to understand it. If not, take a bit longer. If not > at all, try to study something other than math. 5. I think you should update your knowledge about logical reasoning. 6. You shall refocus on a proof of (*). 7. Does the moon exist if Germany is not a monarchy? F. N. -- xyz
From: mueckenh on 19 Dec 2006 11:20 Dik T. Winter schrieb: > In article <1166495541.239288.43640(a)48g2000cwx.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > > Dik T. Winter wrote: > > > In article <1166474763.304897.177520(a)80g2000cwy.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > > > ... > > > > > That is only the case in finite trees. > > > > > It fails miserably in infinite binary trees in which no path has a last > > > > > node or last edge. > > > > > > > > Do you agree that the number of edges is the upper bound of the number > > > > of paths but disagree that the edges are countable? > > > > > > It is the case in finite trees, but that makes it not true for infinite > > > trees. > > > > So for finte trees there are more edges than paths but for infinite > > trees there are more paths than edges? > > Indeed. LOL. > > So for finte trees there are more edges than paths but for infinite > > trees there are more paths than edges? > > Right. It is taking conclusions about the infinite from the finite cases > when you think it is otherwise. Perhaps we cannot determine the limit of the series 1 + 1/2 + 1/4 + ... unless we admit some sophisticated calculation of limits. But without any such sophistication we can be sure the sum of arbitrarily many terms of that series will be less than 28.50. And you assert that it is greater than 100! That is what you call mathematics. But it is simply a wrong application of human brain. Regards, WM
From: mueckenh on 19 Dec 2006 11:22
William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > Albrecht wrote: > > > > > > but in the same time |N isn't complete > > > > since |N is bijectable to the diagonal and the diagonal is never > > > > complete since there is no complete line (number) which covers the > > > > diagonal. > > > > > > No, the claim is that there is no line that covers the diagonal. > > > > Which results in the conclusion that there is no diagonal. > > No. Only properties of the diagonal that you > have accepted are used. > The conclusion is that there is no line. Every element of the diagonal is n some line. Every element of the diagonal is a finite natural number, by definition. Up to the finite natural number n: Every element =< n of the diagonal is contained in line n. Induction shows this is valid for any finite natural number. There are only finite numbers in the diagonal. QED. Regards, WM |