Cantor Confusion
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From: Dik T. Winter on 23 Dec 2006 19:03 In article <1166908335.956982.51170(a)h40g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > The sum 1 + 1/2 + 1/4 + ... has no last term. Nevertheless we know, > > > even in ZFC, that it is larger than 1 and less than 10. That has > > > significance to our question. The series exactly maps the infinity of > > > the paths. > > > > The sum as stated has *no* meaning in mathematics unless you are using > > limits > > The bijection {1,2,3,...n} <--> {2,4,6,...2n} is a bijection for > finite sets. In fact, it is false for infinite sets. > > The sum 1 + 1/2 + 1/4 + ..., however, is larger than 1, with any > definition of limit. > > > In the previous messages (and this one) you state something akin to: > > lim{n->oo} ((# edges@n)/(# paths@n)) = > > lim{n->oo} (# edges@n) / lim{n->oo} (# paths@n). > > The first limit is indeed 2. But that does not mean that both limits > > on the right hand side can be compared with that. The two limits do > > not exist. So the theorem "the limit of a quotient is the quotient of > > the limits" can not be applied because that has only been proven for > > the finite case. > > The sum 1 + 1/2 + 1/4 + ... > 1 is sufficient for my proof. No > division, no limits will save you from going down. I now understand what you were trying to do. But in the complete tree the parts of edges that are assigned to an infinite path are not 1, 1/2, 1/4 etc. There is *no* edge that is completely assigned to a path, and there are also not 1/2, 1/4, etc. edges assigned to a path. So that sum is irrelevant to the infinite tree. Note that in each finite tree, the complete edge assigned to a path is the last edge on the path. As there is no last edge in an infinite path, this does not hold in the infinite tree. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Newberry on 23 Dec 2006 19:47 Dik T. Winter wrote: > In article <1166845904.426550.122020(a)48g2000cwx.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > > stephen(a)nomail.com wrote: > ... > > > > What is wrong with WM's mapping? You divide each edge into two halves > > > > and pass one half to each branch. Then you divide the passed half again > > > > and again. > > > > > > So which edge is mapped to the path that always goes left? > > > Remember, you need to uniquely map each path to an edge. > > > > No, I don't. What I am showing here is that each path will accumulate > > the weight of two edges as it approaches infinity. > > No, you do not. In the finite case you do, approximately. In the > infinite case you do not because *all* parts you pass to a path are > reduced to size 0. Consider what we pass on to the paths that go > left only. Look at the following: > level 1: 1 edge at level 1 > level 2: 1/2 edge at level 1 + 1 edge at level 2 > level 3: 1/4 edge at level 1+ 1/2 edge at level 2 + 1 edge at level 3 > etc. for each finite n, but what about the infinite case? > level 'oo': 1/oo edge at level 1 + 1/oo edge at level 2 + ... What is level oo? The levels are indexed by natural numbers. > Where do you see the 2 coming up? It is certainly *not* the series > 1 + 1/2 + 1/4 + ..., because there is *no* edge that is passed in > full, there is also *no* edge of which 1/2 is passed, etc. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 23 Dec 2006 19:55 In article <1166907306.745536.315650(a)h40g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1166737132.026953.46380(a)48g2000cwx.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > Indeed. No axiom is necessarily true. > > > > > > How then should the truth of AC be proved? > > > > One does not prove axioms, one either assumes them or does not assume > > them. > > You like to snip in such a way that a wrong meaning of my words becomes > manifest? Since your words are full of wrong meanings, anything that reveals this truth is to be supported. > ============= > >> The sum 1 + 1/2 + 1/4 + ... has no last term. > > >It is not a sum, but is often used to represent an infinite sequence of > >partial sums. Such a sequence may have a defined limit value, but that > >limit is not "the sum" except by abuse of language. > > Whatever you may call it. The value < 1 required by set theory can only > be attached to it by total abuse of human brain Set theory alone does not attach a value to it at all. It is with the aid of analysis that 1 + 1/2 + 1.4 + ... may be associated with a limiting value, and that value is in analysis is not "< 1" as you falsely imply. But also that value is irrelevant in judging the relative numbers of edges and paths in infinite trees. > > >Except that the only demonstration of such "inacceptability" would be > >the discovery of some internal contradiction within ZFC or NBG. > > No. The necessary inequality in ZFC 1 + 1/2 + 1/4 < 1 is enough. As that inequality does not occur anywhere except in WM's delusions, it is not relevant to the consistency of ZFC not the number of paths in an infinite tree, > > >And the only conflicts WM can up with are with statements outside of ZFC > >and NBG, which are irrelevant to what occurs within them. > > Everything that can be said against ZFC is irrelevant, by definition. Unless one can show internal inconsistencies in an axiom system, one cannot show it to be inconsistent at all. That is the nature of axiom systems. What WM is really trying to say is that there is some way in which it does not conform to his views of what should be allowed in mathematics. But that is an entirely different issue.
From: Virgil on 23 Dec 2006 20:11 In article <1166907551.057290.90300(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Are you unable to see that the formal use of fractions like |N|/|R| < 1 > > > is completely irrelevant but only a brief way to express |N| < |R| ? > > > > By definition, |N| < |R| means no more than that there are injections > > from set N to set R but none from set R to set N. > > > > How is E(n) < P(n) relevant to that definition? > > If you agree that the the inequality aleph_0 < 2^aleph_0 is false or > useless, then I have reached one of my aims. I agree to neither. > But if you do not, then > you must agree that the ">" symbol can be used in another framework > too. (At least if you applied some logic.) When one allows that "<" may have meaning, one has not automatically accepted the same for ">". But I do allow that ">" may have meaning in some contexts, it is merely that I do not always allow that WM's attempts to use it are meaningful. > > > There is no line containing every node. In my model, in which the diagonal is any full path in an infinite tree, and the lines are the finite subpaths starting from root node to end at some finite node of that path, this is easily seen. > > Which number is missing in every line of the EIT? Quantifier dyslexia! No one claims that. If L = set of lines, and N = set of numbers (A x in L) ((E n in N) (n not in x)) This does NOT require that (E n in N)((A x in L)(n not in x)) And in ZFC the former is true when the latter is false. > Which number is missing in line A which is not missing in line B while > line B contains a number that is missing in line A? Irrelevant! > > > Note that WM has so far carefully avoided responding to this refutation. > > Any infinite path is a diagonal for some EIT. > > The finite subpaths staring at the root nod and ending at > > a node of the given path are the edges. > > Given any node there is a line containing it > > BUT > > There is no line containing every node. > > Which node is missing in every line? Only WM says anything like that! (A x in L) ((E n in N) (n not in x)) but not (E n in N)((A x in L)(n not in x)) > > =================================== > > >> It is a number which is at least as large as the largest existing > >> natural number. > > > And must be larger than its successor. > > No, it is always the successor which is the largest number. But as every successor is also a predecessor of another successor, that can never happen.
From: Virgil on 23 Dec 2006 20:24
In article <1166907866.704649.25090(a)h40g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > > > > > > > I can calculate 1 + 1/2. > > > > > > > > Good for you! > > > > > > Bad for you, if you can't. > > Can't you? > > > > I know that 2/3 < 1 is a convenient way to express 2 < 3. > > > > For heaven's sake. "2/3 < 1" is not just "a convenient way" of > > expressing "2 < 3". > > It is *also*, among others, a convenient way of expressing "2 < 3 > > > > > E(n)/P(n) -> 2 is a feature only of finite n. |E(n)| < |P(n)| is a > > feature only of infinite n. /Many/ things that can be said in the > > finite case are not true in the infinite case; > > But is *not* among those. No one but WM has ever claimed " 1 + 1/2 + 1/4 + ... < 1" Why he is so hung up on such an obvious falsehood is a puzzlement. Unless He is assuming that the limit of a quotient must equal the quotient of two limits when the latter limits do not exist. This non-theorem is non-true in ZFC. > > > for example E(n) is > > finite for natural n, and not finite for infinite n. > > For example {1,2,3,...n} <--> {2,4,6,...2n} is a bijection for finite > sets. It is obviously false for infinite sets. As neither of the sets, as described, can be infinite, it has nothing to do with infinite sets. > > It is when you try to argue "lim n-> oo E(n)/P(n) = 2 contradicts > > E(n)/P(n) < 1 in the infinite case" that your confusing notation > > "looks" like a contradiction (to you; to me it looks like nonsense). > > Then you really do accept 1 + 1/2 + 1/4 + ... < 1 ? What brings you to the illogical conclusion that the "limit" has anything to do with anything? If n means card(N) in "E(n)/P(n)" then E(n) and P(N), both being infinite, make that quotient undefined. What WM is really trying to claim is that "lim n-> oo E(n)/P(n) = (lim n-> oo E(n))\("lim n-> oo P(n))" but at least 2 of those 3 limits do not exist, so it is a false equality. |