Cantor Confusion
in [Math]
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From: Virgil on 10 Feb 2007 17:33 In article <1171114429.303660.273590(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 9 Feb., 16:22, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > On Feb 9, 4:15 am, mueck...(a)rz.fh-augsburg.de wrote: > > > And at this point you acknowlege that is it possible > > to define the sparrow of E. So the sparrow > > of E exists. > > Do you think that everything exists which I acknowledge? Since WM does not acknowledge several things which do exist and already does acknowledge things which do not, there seems to be little correlation between what WM acknowledges and what exists. > > > > If we decide to call the sparrow of E a number, then it > > is not a natural number > > But it does not mean that this sparrow is alive. > If we decide to call a number between 1 and 2 a natural number, then > this is a wrong definition. That depends on whose to be master, that's all! > > > > No statement you make about things that are true > > of every set with finite cardinality, or things that > > are true for every natural number, can be used > > to show something about the sparrow of E. > > The set E is not a set with finite cardinality > > and the sparrow of E is not a natural number. > > The fact that E is composed of sets with finite > > cardinality does not mean that E is a set > > with finite cardinality. > > The finite cardinality has been proved by complete induction. False! WM has yet to prove anything to the satisfaction of anyone other then himself. And until others accept such a proposed proof, it is no more than one person's opinion. > > > The fact that the cardinality > > of E can be seen as the limit of natural number > > does not mean that the cardinality of E must > > have the same properties as the natural > > numbers. A limit of a sequence does not > > have to have the same properties as the elements > > of a sequence. > > A limit of a sequence (a_n) has to have the Cauchy property. omega - n > = omega does not satisfy it. Firstly, "omega - n = omega" is neither a seqeunce nor the limit of one, so that WM is, as usual, both wrong and irreelvant. Secondly, a limit, or union, of a sequence of sets, need not have any Cauchy property as a set. It is only sequences of real numbers for which that would be relevant. > > > > > > > > > > Extending the concept of cardinality to include > > > > potentially infinite sets does not lead to > > > > a contradiction. > > > > > Unless you say that it is a number larger than any natural number. What is contradictory about having a number which is not a natural that is "larger than" any natural? That is no more contradictory that having a number which is not a negative larger than every negative number. > > > > No. If you extend the concept of cardinality to potentially > > infinite sets, then the cardinality of a potentially infinite set > > is not a natural number, > > It is not a number and cannot be a number. Who gave WM the right to determine what may be called numbers and what may not be called numbers? That is determined by general consensus, and the general consensus is that WM is wrong! > It can only be the property > that the natural number which is the cardinality of the set in present > state can grow. WM has a remarkably poor understanding of what "sets" or "cardinality" are about if he imagines that cardinality applies to sets which are allowed to be growing. > But the property "can grow" is as little a number as > the property can eat hot dogs or can drive a red car is a number, let > alone an infiite number. "Can grow" is irrelevant to set theory, including cardinality. It is a delusion that WM does not seem able to grow out of.
From: Virgil on 10 Feb 2007 17:42 In article <1171114705.923162.72680(a)v45g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 9 Feb., 21:26, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > > > > Reality. > > > > What reality? Empirically testable physical reality? A mathematical > > statement is not of that kind. > > Every mathematical statement is of that kind. That Wm thinks so is convincing evidence of his inability to deal with mathematics honestly or fairly. > You would recognize this > (if you could, but you couldn't) if all reality ceased to exist. If all reality ceased to exist, no one wold be around to realize anything. Not even WM. > No mathematics would remain. Actually, the mathematics, being quite independent of reality, could remain, it is only the mathematicians, and anti-mathematicians like WM, who would no longer remain. > Yes, to grasp these facts, without such a > bad experience, i.e., during the continued existence of reality, one > needs some deeper thinking than is required to understand the meaning > of some crazy axioms. Then WM should try to think a bit more deeply so as to understand the craziness of his own axioms, the ones he denies he has but still beleives.
From: Lester Zick on 10 Feb 2007 17:45 On Sat, 10 Feb 2007 15:36:45 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sat, 10 Feb 2007 12:06:02 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >mueckenh(a)rz.fh-augsburg.de wrote: >> >> On 9 Feb., 21:26, "MoeBlee" <jazzm...(a)hotmail.com> wrote: >> >> >> >> > > Reality. >> >> > >> >> > What reality? Empirically testable physical reality? A mathematical >> >> > statement is not of that kind. >> >> >> >> Every mathematical statement is of that kind. You would recognize this >> >> (if you could, but you couldn't) if all reality ceased to exist. >> > >> >Are all statements (not just mathematical ones) about reality? > >Something from the "Lord of the Rings" trilogy. Not that I'd necessarily disagree but I just wonder if you could be a little more specific? ~v~~
From: Virgil on 10 Feb 2007 17:49 In article <1171115167.832067.240100(a)j27g2000cwj.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 10 Feb., 12:13, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: > > > > >> > Instead of a path P consider the set S of nodes K which belong to a > > >> > path P. Do your calculation and arguing. Then substitute P for S to > > >> > have a brief notation. > > > > >> One cannot determine merely from a set of nodes for a given tree > > >> whether that set of nodes is or is not a path. > > > > > > One can. > > > > Then please do so: > > > > given tree T := { a, b, c, d, e, f, g } > > given set of nodes S := { a, b, c } > > > > Tell us whether S is a path in T. And please explain that. > > > > > One has the coordinates of the node. > > > > The what? > > The co-ordinates: The first coordinate is the number n of the level, > the second coordinate m is the number within the level, counted from > the left, for instance: (n,m) = (3,4) means the node in level 3 which > is the 4th from the left edge of the tree. > > Regards, WM What if one prefers a quite different coordinate system, such as the sequence of left or right branchings from the root node to the node in question? There is no required coordinate system, and those who do not like WM's are not required to acknowledge it. And, indeed, given the set of nodes and set of edges as required by the standard and original definitions of trees, no coordinate system is requires at all.
From: Virgil on 10 Feb 2007 17:58
In article <1171129864.796664.8490(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 10 Feb., 12:13, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: > > > > >> > Instead of a path P consider the set S of nodes K which belong to a > > >> > path P. Do your calculation and arguing. Then substitute P for S to > > >> > have a brief notation. > > > > >> One cannot determine merely from a set of nodes for a given tree > > >> whether that set of nodes is or is not a path. > > > > > > One can. > > > > Then please do so: > > > > given tree T := { a, b, c, d, e, f, g } > > given set of nodes S := { a, b, c } > > > > Tell us whether S is a path in T. And please explain that. > > Pardon, I overlooked your first question. > > The tree T(2) in my notation can be given as a chain by > > a > bc > gfed > > abc is the subtree T(1) with only (the root node and) one level. It is > not a path. > > > Here are some further representations of the tree T(2): > > 0. > | \ > 0 1 > | \ | \ > 0101 > > briefly (edges are not necessary) > > 0. > 0 1 > 0101 > > > (0,1) > (1,1) (1,2) > (2,1) (2,2) (2,3) (2,4) > > The last is the coordinate representation. I think it is obvious which > subsets are paths. Then give an "obvious" test rule to distinguish between sets of nodes which are paths and set of nodes which are not paths. > > One of the paths in the infinite tree is the set of nodes p(oo) = {(n, > 1)| n in N or n = 0}. It can also be written as a,b,g, ... or as > 0.000.... In no case the order property need be mentioned. If 0.000... is one of the paths in WM's infinite tree, should not every sequence with any pattern of "0"s or "1"s also be a path in WM's infinite tree? And are the number of such sequences not uncountable? Cantor has proven the set of them uncountable! It appears that WM loses again! |