Cantor Confusion
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From: Virgil on 5 Jan 2007 16:13 In article <1167999736.624629.206820(a)s80g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > And my question is: Do these two trees, namely the complete tree and > > > the union of all rational trees differ such that the one has edges or > > > nodes which are missing in the other? > > > > No. > > > > But the distinction is apparently difficult (although about first year > > at University for mathematics). The set of terminating binary expansions > > is countable, the set of non-terminating binary expansions is no > > countable. (You may replace terminating binary expansions with binary > > expansions terminating with either a continuous stream of 0'z or of 1's.) > > But these streams are *not* present in all paths of the union of all > rational trees! No one, except possibly WM, is claiming that an infinite binary tree is the same as the union of all binary rational, and therefore finite, trees. >That has been overlooked, as it appears, even in the > last years of university mathematics - in the last 130 years. Possibly because it is irrelevant. > > Can you really believe that a thinking brain will accept your assertion > that two absolutely identical systems of nodes and edges will supply > different systems of paths, i.e., strings of nodes and edges? No one is requiring any such thing. Those who have thinking brains are capable of distinguishing between the union of infinitely many finite trees and a single infinite tree. > > > > > > You are wrong. sqrt(2) has a pretty good representation: "sqrt(2)". > > > > > > That is a name. Name ist Schall und Rauch. If we cannot refer to a number by its name, we cannot refer to numbers at all.
From: Virgil on 5 Jan 2007 16:15 In article <1167999883.980083.264720(a)q40g2000cwq.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David R Tribble schrieb: > > >> David R Tribble schrieb: > > >> >> Well, now I'm confused. Could you provide an example of a natural > > >> >> number that does not exist? > > >> > > > >> > > >> mueckenh wrote: > > >> > Take the first 10^100 digits of pi (if you can - but you cannot). > > >> > It is impossible to bring this number to existence in the whole > > >> > universe. > > >> > > >> Can you "bring into existence" the number 1? > > > > > > Here it is: > > > > > > . > > > > [X] a dot > > One clear representation of number 1, sufficient to identify it. Not > only a name. Do unnamable numbers have any existence at all? What numbers are ever referenced in serious mathematics except by being named?
From: Virgil on 5 Jan 2007 16:23 In article <1168000528.484900.250330(a)s80g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > I do not assume that 0.111... exists. I only assume that > > given that 0.111...1 exists you can show that 0.111...11 exists. > > > That position is correct. It is called potential infinity. Except that it requires that as soon as any natural is named or otherwise referenced, its successor already exists. > But you always assert that the the complete diagonal exists, i.e., a > diagonal which could not be extended. Then you must also accept a line > system which cannot be extendend. Maybe you feel compelled to accept that, but WE don't have to accept any such thing. In OUR math, every "line" ends, but the collection of lines, and therefore the collection of line endings, does not have an "end" or last member.
From: Virgil on 5 Jan 2007 16:28 In article <1168000832.112048.132050(a)38g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > >Otherwise I had to assume that the union of all finite binary > > > trees is identical to the complete binary tree, as far as nodes and > > > egdes are concerned, but that the trees are not identical as far as > > > paths are concerned. > > > > What is wrong with assuming what is easily provable? As every edge and > > node is a member of some finite path of some finite tree, each must be > > in the union. > > So it is. > > > But no infinite path is a member of any finite tree, so > > none can be members of a union of finite trees. > > So where are the nodes and edges which make the infinite paths longer > than any finite path? If you don't know, no one can tell you. > > > > > I cannot accept ghosts in mathematics, but, of course, I cannot prove > > > that they do not exist. So we are finshed with this topic. > > > > WE, on the other hand, can prove that what WM regards as ghosts do exist > > in several versions of set theory. > > That makes these versions of set theory vakueless. Only to WM. They are quite valued by many others. > > >As WM has no specific version of his > > own, he effectively has none at all. > > There is no version of infinity other than by belief in ghosts (lacking > mathematical spirit) I am not expecting WM's "system" to include what he does not believe in, but until he has an explicitly stated axiom system which others can check for internal consistency, he is not in a position to criticize those who do have such systems.
From: Virgil on 5 Jan 2007 16:32
In article <1168001123.255699.292040(a)11g2000cwr.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > > > exist. > > > > They do in ZFC or NBG. > > Is that the theory which requires different paths from identical nodes > in the binary tree? I do not know what sort of trees WM is dreaming about, but in my infinite binary trees, at each node there are infinitely many paths branching left and infinitely many others branching right. > > Virgil, if you believe that stuff, then any discussion with you will be > in vain, in particular in fields where the contradictions are not so > obvious. Any contradictions that WM believes he sees in my trees exist purely in his own head and nowhere else. |