Cantor Confusion
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From: mueckenh on 7 Feb 2007 07:36 On 6 Feb., 13:28, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > Let E be the sequence of the sets F(e_i). There are two statements. > > i. For each of the sets F(e_i) in the sequence E, there is > an element of F(e_i) which is greater than the cardinal > number of F(e_i) > > ii There an element of one of the F(e) that is greater > than the cardinal number of E. > > i is true, ii is false, E is a counterexample. E is not an example, because it is self contradictory. The potentially infinite set of even numbers is *constructed* by its segments {2,4,6,...,2n} Every time we increase n by 1 we increase 2n by 2. This cannot be avoided. Therefore it is impossible to have for finite natural numbers lim[n-->oo] |{2,4,6,...,2n}| > 2n or lim[n-->oo] 1/2 > 1. But there are no other than finite natural numbers. This simple truth shows that your belief in the actual existence of E is completely unjustified and wrong - whether there is an axiom saying so or not. Regards, WM
From: mueckenh on 7 Feb 2007 07:40 On 6 Feb., 13:47, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > It can be proven. {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } > > contains {p(0), p(1), p(2), ... } > > What does contain mean? The tree is a set of nodes. It contains paths as subsets. The special meaning of "contain" is irrelevant as long as we refer to a unique meaning, i.e., either: something which is contained in the tree is really in the tree, or: something which is contained in the tree does belong to the tree as a supremum. > > > Correct. And a set of nodes can be a path. > > No. A paths *is* > > a) a sequence of nodes, or > b) a sequence of edges. > Yes to a and b. But not every set of nodes or edges is a path. > > You have simply stated the Mückenheim axiom > > X is not finite -> there must be an x in X which is infinite > > which is alas not part of ZFC or any other modern set theory. I did not start off with this assumption. I showed: 1) The union of finite trees contains the union of finite paths. 2) The union of finite trees is the whole tree. 3) The whole tree contains all paths. 4) The union of finite paths contains all paths. 5) This union is countable. Regards, WM
From: mueckenh on 7 Feb 2007 07:42 On 6 Feb., 20:46, Virgil <vir...(a)comcast.net> wrote: > WM must be ghost ridden to see them where they are not. > > The union over {{2,3},{5,7}} is {2,3,5,7}, which is not a member of > {{2,3},{5,7}}. Again such a ridiculous example. The union over {{1}, {1,2}} is a member of {{1}, {1,2}} . The union over all paths is a member of all paths. > > > Which requires a path with no end. > > > No. > > Then name that alleged end of the union of all paths! I do not assert that acual infinity exists. Therefore I need not name an actually infinite path. > > > In particular because there cannot be such a path. > > In mathematics there can be such a path, regardless of WM's > misrepresentations. > > > But if you > > assume its existence, then you see that the reals are countable or > > that identical nodes yield different path systems, an idea which > > presumably only you can utter. > > I do not assume it, I prove it, It is nonsense and remains nonsense to assume that a given tree has different path systems according to... according to what? According to your emotional condition? And if you prove this nonsense then you have found a contradiction in set theory. > and I see that the reals are uncountable > and WM's pseudo-unions of trees are mathematically corrupt and > self-contradictory, and those are ideas that any sane person can utter > regardless of WM's presumption What defines the different path systems in the tree with fixed nodes and edges? Can the system increase and decrease? How can one influence its behaviour? Regards, WM
From: mueckenh on 7 Feb 2007 07:44 On 6 Feb., 20:48, Virgil <vir...(a)comcast.net> wrote: > In article <1170754974.135681.22...(a)p10g2000cwp.googlegroups.com>, > > > > The simplest reason is that omega - n = omega for all n in N. > > > > Where did you get that from? Reference? EB? > > > You could even figure it out by yourself. > > If one needs to adopt WM's crazy assumptions in order to come to WM's > crazy conclusions, we are all better off without them. Both his > assumptions and his conclusions Excuse me, these were Cantor's assumptions and conclusions. Cantor, Collected Works p. 323: If a and b are two ordinals with a < b, then there exists an ordinal which we call b - a ... You see what value your insults have. (Cantor was used to insults, I am used to insults. No problem.) Regards, WM
From: mueckenh on 7 Feb 2007 07:46
On 6 Feb., 20:58, Virgil <vir...(a)comcast.net> wrote: > In article <1170755634.350929.67...(a)s48g2000cws.googlegroups.com>, > > mueck...(a)rz.fh-augsburg.de wrote: > > The existence of the empty set is not at all guaranteed. > > It is in ZFC. > > ZFC requires at least one set to exist, and also requires existence of a > subset of that set which does not contain any of the members of that set. > > > There is an > > axiom which requires the existsence of the non existing and seems to > > make some people happy > > If there is such an axiom anywhere, it is only an axiom in WM's system, > not in anyone else's. > > > (like the axiom which requires the finity of > > the infinite). > > Any such axiom exists only in WM's system, and not in anyone else's. > Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations of Set Theory", 2nd edn., North Holland, Amsterdam (1984): "Intuitionists reject the very notion of of an arbitrary sequence of integers, as denoting something finished and definite as illegitimate. Such a sequence is considered to be a growing object only and not a finished one." Who considers it a finished one? Regards, WM |