Cantor Confusion
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From: Virgil on 14 Oct 2006 18:11 In article <1160852959.173914.326370(a)e3g2000cwe.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Brouwer's Continuity Theorem is cosmologically valid. Not in my cosmos.
From: David Marcus on 14 Oct 2006 18:12 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > Please state an internal contradiction of set theory. Please use the > > > > standard language of set theory/mathematics so that we can understand > > > > what the contradiction is without needing to ask what all the words > > > > mean. > > > > > > Good heavens, there are so many. Where shall I start with? > > > > > 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. > > > > Are you using "relation" in its mathematical sense? > > Of course. But instead of whole elements, I consider fractions. That is > new but neither undefined nor wrong. > > > > Please define your terms "half an edge" and "inherited". > > I can't believe that you are unable to understand what "half" or > "inherited" means. > I rather believe you don't want to understand it. Therefore an > explanation will not help much. In standard terminology, a "relation between paths and edges" means a set of ordered pairs where the first element of a pair is a path and the second is an edge. Is this what you meant? I am at a loss as to how "half an edge" can be "inherited" by a path. As far as I know, a path is a sequence of edges. Is this what you mean by "path"? Is edge a the line connecting 0 in the first row to 0 in the second row? Or, is it the line connecting 0 in the first row to 1 in the second row? Or, is it something else? -- David Marcus
From: Virgil on 14 Oct 2006 18:15 In article <1160853952.079396.236640(a)f16g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > You forget that _all_ applied mathematics is an approximation. You can > employ the above only in the domain where it is (always approximately) > valid: Newtonian mechanics. But even relativistic mechanics is _never_ > exact, though it may be beter with high velocities. Thus the keywords > are approximation and uncertainity, everywhere where Applied rules the > roast. And that's precisely the point where OUR "paradoxes" disappear. But in mathematics, "applied" does not rule the roost. There is math which is, as yet, incapable of application. If no such math had ever existed, much, if not most, of present day physics would not exist.
From: Virgil on 14 Oct 2006 18:18 In article <1160855574.230559.141770(a)e3g2000cwe.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Dik T. Winter schreef: > > > In article <b2f47$452f51eb$82a1e228$2726(a)news2.tudelft.nl> > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > ... > > > b(t) diverges at noon. Thus b(noon) is undefined. Thus t is not time. > > > > > > It's impossible in a renormalized mathematics that limits are different > > > from the actual values of functions at that place. > > > > "Diverges at noon" is strange wording in my opinion. I would say > > b(t) diverges when going to noon. And so the limit does not exist. > > Yes. That's a better wording perhaps. > > > But this is not in contradiction with b(noon) = 0. > > NO with contemporary mathematics. But with my renormalized mathematics: As the problem, as stated, does not require, or even allow, "renormalization", HdB is not working on the original problem but on one of his own creation, which is irrelevant to the original.
From: Virgil on 14 Oct 2006 18:19
In article <1160856043.795135.198610(a)h48g2000cwc.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Dik T. Winter schreef: > > > In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl> > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > > > Set Theory is simply not very useful. > > > > Oh. So you think that Banach spaces are not very useful? You think that > > a book like "The Algebraic Eigenvalue Problem" is not very useful? You > > may note that both are heavily based on set theory. > > Think I could rewrite the relevant stuff in those books without using > any set theory. I'm not actually going to do it, though. (BTW, I find > Banach Spaces not very useful either) > > Han de Bruijn That's OK. we do not find HdB very useful, either. |