Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 29 Dec 2006 17:02 Virgil schrieb: > > > ZFC or NBG set theory might corrupt the peculiar faiths of those who > > > believe as WM does, but is perfectly consistent with standard logic. > > > > You see that your standard logic enforces the opinion that an existing > > individual entity does not exist as an individual entity. > > Which "individual entity" is that? > That one which distinguishes a path from all its co-paths. > > > > >The same " limit" argument will conclude the in a tree in which no path > > > has a terminal node, every path has a terminal node. > > > > No, there are meaningful limits and meaningless limits. > > All the limits involved here are equally meaningless. The limit n --> oo covering all natural numbers is the only meaningful limit in mathematics. > > > There is no such thing as a single "separated path" > > > > But the cardinal number of all these not being no-things is 2^aleph_0? > > What nothings? There are paths, but in an infinite tree, no path can be > separated from all other paths Every (existing) path is separated from all other paths but no path can be separated from all other paths. > by any one node or edge. Not by one node or edge? By what else? > > > > ============================= > > > > > These things are all easily seen if one notes that any infinite path is > > > an infinite sequence of left/right branchings. > > > > But no infinite path exists individually? > > If that is how you choose to misunderstand things! Either there is an edge which separates a path from all other paths Or it is impossible to separate a path from all other paths. > > > > > For any natural n, cutting off the first n branchings of any (infinite) > > > path leaves an (infinite) path, and for fixed n, the set of all such > > > truncated paths is identical to the original set of all paths. > > > > And that is so, although no original path does exist. > > Who says no original path exists? They all exist but are not "separated" > by the means you claim. By what is a set which only exists of edges separated from another set which also only exists of edges? =================== Virgil: It is not enough or WM would have proceeded to do the proof. What Wm continually ignores is that in an infinite tree, through each edge pass infinitely many paths, indeed, uncountably many, so that each new edge produces a new infinity of paths. WM: So each irrational number is in fact an uncountable set of numbers? Virgil: It is a delusion that there are only two paths, unless those two edges are both terminal edges. If neither edge is a terminal edge, then there are at least 4 paths, 2 through each, for longer paths more edges, and for endless paths uncountably many edges. WM: Call them bunches of paths. So any irrational number is represented by a bunch of paths? Virgil: In an infinite tree one is merely "personalizing" an whole uncountably infinite set of paths, not a single path. WM: So any irrational number is represented by a real interval? Virgil: For every real in [0,1], there is a path WM: One path? How does the path of 1/sqrt(2) differ from every other path? Virgil: Except that WM is incapable of producing that alleged proof, while others are quite capable of providing proofs that his claim is false, at least in ZFC and NBG. WM: These proofs, if not invalid, would at most prove the inconsistency of ZFC. Regards, WM
From: mueckenh on 29 Dec 2006 17:16 Virgil schrieb: > In article <1167392633.967611.252860(a)n51g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > Apparently your limits work only for real numbers in lists but no for > > > > real numbers in trees. > > > > > > Oh, they do. If the trees are properly defined. > > > > Imagine a tree which contains only paths of rational numbers in > > infinite representation, i.e., ending with a period of 000... or > > 111.... > > > > The first rational tree contains only the paths 0.000... and 0.111... > > > > 0. > > / \ > > 0 1 > > / \ > > 0 1 > > ............. > > > > The second tree contains the paths 0.000..., 0.0111..., 0.1000..., and > > 0.111... > > > > and so on until all rational numbers are represented. > > > > Take the union of all these rational trees. It contains all paths > > representing rational numbers. > > > > What distinguishes this union of rational trees from the complete tree? > > Nothing. > > Except that it is easy to construct a path in the complete tree not in > any member of the union, and therefore not in the union itself. Fine. Tell me the node or edge which differs from the nodes or edges of the union. > > x = 0.101001000100001000001.... cannot be in any member of that union. > Alternately, x = sum{n=1..oo} 1/(2 ^ ((n^2+n)/1) I know about your fairy tails, but these trees are constructed by edges and nodes, rigorously. Something that is claimed to exist in the complete tree but not in the union of all rational trees must be distinguished by a node or edge which is not in the latter, but in the former. We will do strict mathematics of nodes and edges here and not matheology of possibly imaginable mythes. Point to the node supporting the difference or withdraw your dream. > > > There is no node and no edge of the complete tree which is not > > in the union of all rational trees. > > But there are lots of paths which are not in that union. Same edges, different paths. But you do not begin to doubt ZFC? You would rather prefer to doubt your own existence. > > E.g., given any f:N -> N which is a strictly increasing polynomial > function of degree greater than 1, then y = sum 1/2^f(n) is not in your > union. That means, it is not anywhere. > > > > As a path is an ordered set of > > edegs (and nodes), two paths are not different unless they differ by an > > edge (or node). Hence, there is no path in the complete tree which is > > not in the union of all rational trees. > > WRONG! See above for specific examples of transcendental numbers > corresponding to paths not in your tree. Examples of dreams are not paths in a real tree, not even in a Christmas tree. Regards, WM
From: Virgil on 29 Dec 2006 18:37 In article <1167428956.928641.196030(a)v33g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > You don't know or avoid to confess that the number of digits > of their rational approximations is limited? But that doesn't change > that fact. The number of digits in the decimal approximations of any rational not expressible as a fraction with power of 10 as denominator, is equally limited. Does that mean that those rationals do not exist?
From: Virgil on 29 Dec 2006 18:41 In article <1167429117.398074.270070(a)a3g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > It is possible to show: > When these numbers will exist, the due maximum will exist. > That's enough. As soon as a maximum is "shown", it is also shown that it is not a maximum. > > > L_D does not only have to include numbers that have been shown > > to exist, it must include all numbers that it is possible to show > > exist. > > Why? How should we know which can possibly exist? Those of us with a bit more wit than WM know how. If it is possible for n to exist, then it is possible for n+1 to exist, so that there cannot be any finite limit on what can possibly exist.
From: Virgil on 29 Dec 2006 18:46
In article <1167429289.476619.27000(a)48g2000cwx.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Newberry schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > Therefore: If you follow some path you will see that whenever it > > > > > separates itself from another one, this happens by two edges .- one > > > > > edge for the path, and the other edge for the other path. This > > > > > process > > > > > repeats and repeats without end. Nothing else happens. From the > > > > > unavoidable and "inseparable" connection of separation and > > > > > appearance > > > > > of another edge we can conclude that every path which can be > > > > > distinguished from another path runs through an edge which does not > > > > > belong to the other path; call it "personalization of an edge. > > > > > Therefore distinguishability of paths and personalization of edges > > > > > are > > > > > unavoidably connected. > > > > > > > > For each two paths you can identify an edge where they separate. But > > > > there > > > > is *no* edge that separates (for instance) the path leading to 1/3 from > > > > all other paths. And through each and every edge run infinitely many > > > > paths. > > > > > > Yes. That shows that there are no infinite sequences of edges (or > > > digits) which individually represent nunmbers like 1/3 or ideas like > > > irratinal numbers. So that in WM's philosophy, 1/3 does not exist but 1/2 and 1/4 do? > > > > But there are algorithms that tell at EACH level whether to go left or > > right. > > And there are paths of other numbers which, up to EACH level, behave > exactly as the path of the binary representation of 1/3 does. Anything that behaves like 1/3 at EACH level is 1/3. > > Further, the algorithm for 0.010101... works only as long as you can > determine whether the number n of the level is even or odd. All one has to be sure of is that they alternate 0 an 1. |