Cantor Confusion
in [Math]
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From: mueckenh on 10 Feb 2007 08:38 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. No mathematics would 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. Regards, WM
From: mueckenh on 10 Feb 2007 08:46 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
From: William Hughes on 10 Feb 2007 09:31 On Feb 10, 8:33 am, mueck...(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:\ I start by restoring some stuff you snipped H: B: every set of finite even numbers H: contains numbers which are larger than the cardinal H: number of the set. M: This implies that the cardinal nunmber of the set is finite. H: Only if the statement is true. If the statement is H: false for a set K we do not know whether H: -K does not have a cardinal number H: or H: -K has a cardinal number but K does not H: have an element larger than its cardinal H: number. So the statement every set of finite even numbers contains numbers which are larger than the cardinal number of the set. is false. If you want it to be true you need to change "every set" to something like "every set that is complete" or change your mind and say that E is not a set of finite even numbers. More snipping restored H: However, saying that E does not have a cardinal H: does not change the properties of E. E still H: has a sparrow (recall a sparrow is an equivalence H: class under the equivalence relation equitransform H: which generalizes the concept of bijection to include H: potentially infinite sets). This sparrow can be H: compared with other sparrow's including the sparrows H: of finite sets, which are just the finite cardinals. H: There is nothing contradictory about defining H: a sparrow. M: Unless > > > 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? Your use of the term "unless" means that you think there is no contradiction unless something is added. > > > 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. And if we decide to call the sparrow of E a natural number then this is a wrong definition. So " If we decide to call the sparrow of E a number, then it is not a natural number" > > > and this statement is not > > a contradiction. > > It is. > > > > > 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. > No the finite cardinality of the components of E has been proved by induction. E is not one of the components of E, and the fact that the compenents of E have a given property does not mean that E has that property > > 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. Piffle. We are not talking about the convergence of real numbers to a real number. [Note also: it is the sequence that has the Cauchy propery, not the limit] > omega - n > = omega does not satisfy it. > > > > > > > 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. > > > 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. > It can only be the property > that the natural number which is the cardinality of the set in present > state can grow. No. One possible definition for the cardinality of E is the sparrow of E. The sparrow of E is a fixed equivalence class. So the statment [The cardinality of E] can only be the property that the natural number which is the cardinality of the set in present state can grow is false. is false. - William Hughes
From: Franziska Neugebauer on 10 Feb 2007 11:14 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. 1. Nodes do not "know on which "coordinate" they are seated. In your definition of tree even T does not "know" where its nodes seat. 2. You have not answered whether S is a path in T. F. N. -- xyz
From: David Marcus on 10 Feb 2007 12:06
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? > No > mathematics would 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. -- David Marcus |