Cantor Confusion
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From: Ralf Bader on 30 Dec 2006 16:16 mueckenh(a)rz.fh-augsburg.de wrote: > > cbrown(a)cbrownsystems.com schrieb: > >> Have you given up on your "rational relation" proof that |N| = |R|? > > No, why should I? It is correct. (A series with a first but no last > term can be reversed to have a last but no first term - without oosing > its value.) No, because the value is the limit of the sequence of partial sums and this makes no sense if the series is "reversed". > But it is easier to see, and should be visible even for such as you, > that the tree built from the union of all finite trees (with > representations of rational paths) is the same as the complete infinite > tree. The only thing one can see from your idiotic gossip is that you are incredibly stupid.
From: mueckenh on 30 Dec 2006 16:25 cbrown(a)cbrownsystems.com schrieb: > > I did not calculate |N|/|R| but edges per path. > > Wonderful. But what relevance does the calculation of "edges per path" > have to do with the /actual question/: does there exist a surjective > function from the set N to the set R? If there are not less edges than paths, then a surjection is possible, i.e., the possibility of a surjection is proved. It is really the same as with the proof of a well-ordering of the real numbers. And, in fact, if the real numbers could be well-ordered, then the paths could be well-ordered and a bijection between paths and edges (which already are well-ordered) could be completed (because it has been proven that there are not more paths than edges). This is the reason why my proof is important. Do you now see what relevance the calculation of "edges per path" has to do with the /actual question/: does there exist a surjective function from the set N to the set R? > > That is what is meant by "|N| >= |R|". The use of ">=" here has a > /different meaning/ than "1 + 1/2 + ... >= 1", a difference which seems > to elude you. > > In the latter case, we are not asking "is there a surjective function > from 1 + 1/2 + ... onto 1?", because that would be simply nonsense. Only if one does not know the meaning of the arithmetic of rational numbers and limits. Sums of rational numbers are nothing else but sums of integers, given in the unit of the greatest denominator. In case of limits this kind of arithmetic is extended, but the basis is calculating with natural numbers which are to be compared by mappings, i.e., the arithmetic of ordinal numbers. > Similarly, to try to show that |N| >= |R| by calculating the limit of > edges per path is equally nonsense - it requires misreading the meaning > of ">=". You seem to be not very well informed about arithmetic in general, be it rational or ordinal. BTW, have you learned meanwhile that 2^omega is a countable ordinal? Regards, WM
From: mueckenh on 30 Dec 2006 16:27 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > > As not all natural numbers do exist, the set is potentially infinite, > > > > i.e., it is finite. It has a maximum. L_D. Taking this maximum and > > > > adding 1 or takin L_D ^ L_D or so yields another maximum. In any cae > > > > there is a maximum. > > > > > > No. At any time there is a maximum of the numbers that have been > > > shown to exist. > > > > So it is. > > > > > However, there is never a maximum of the numbers > > > that it is possible to show exist. > > > > It is possible to show: > > When these numbers will exist, the due maximum will exist. > > That's enough. > > > > > L_D does not only have to include numbers that have been shown > > > to exist, it must include all numbers that it is possible to show > > > exist. > > > > Why? How should we know which can possibly exist? > > L_D must contain the diagonal. > Thus, L_D must contain any element that > can be shown to exist in the diagonal. It does, after the existence has been shown. Regards, WM
From: mueckenh on 30 Dec 2006 16:30 Virgil schrieb: > > > > > There is no such thing as a single "separated path" > > > > > > > > But the cardinal number of all these not being no-things is 2^aleph_0? > > > > > > What nothings? There are paths, but in an infinite tree, no path can be > > > separated from all other paths > > > > Every (existing) path is separated from all other paths but no path > > can be > > separated from all other paths. > > > > > by any one node or edge. > > > > Not by one node or edge? By what else? > > To separate one path from all others requires an infinite set of edges > or an infinite set of nodes. No finite suffices, as, for any finite set > of edges and/or nodes, if there are any paths through all of them then > there are uncountably many paths through all of them. You said: It is enough to show that for every finite tree, the given path is not in it, in order to show that it is not in the union. I say: It is enough to show that for every finite set of edges, the given path is not individualized, in order to show that it is not individualized in the infinite union. You said: It is enough to show that for every finite tree, the given path is not in it, in order to show that it is not in the union. I say: Is the union of all finite trees different from the complete infinite tree? If yes: Are the irrational numbers a subset of the rational numbers? If no: What is the difference in terms of nodes and edges? > I will but WM keeps going off into his matheology despite his promises. Is the union of all finite trees different from the complete infinite tree? If no: What is the difference in terms of nodes and edges? You said: I see no reason to do either. WM persists in his false claim that a single node or edge is sufficient to separate one infinite path from all others. The truth is that it takes infinitely many edges or nodes to achieve such separations. I say: Infnitely many edges which are shared by two paths or more are not sufficient and are of no value with regards to identify a path. Regards, WM
From: mueckenh on 30 Dec 2006 16:33
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Franziska Neugebauer schrieb: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > Franziska Neugebauer schrieb: > >> >> [...] > >> >> >> >> Linguistic Question: Is it meaningfull to speak of an > >> >> >> >> "approximation" if one denies the existence of the thing > >> >> >> >> which is approximated (the irrational number)? > >> >> >> > > >> >> >> > Not as long as not all deny its existence. > >> >> >> > >> >> >> To me this reads > >> >> >> > >> >> >> As long as all deny the existence of the entity to be > >> >> >> approximated it is not meaningful to speak of an > >> >> >> "approximation". > >> >> >> > >> >> >> Is this correct? > >> >> >> > >> >> >> I do not deny its existence. Do you? What meaning does have an > >> >> >> "approximation" to you if the approximated entity (irrational > >> >> >> number) is supposed to not exist? > >> >> > > >> >> > Ther number does not exist, but the idea does exist. > >> >> > >> >> 1. So you agree that there _exist_ two entities x1, x2 e R which > >> >> solve the equation x^2 = 2? > >> > > >> > Yes. > >> >> > >> >> 2. Is it true that what "we" call an "irrational number" (x1, x2)) > >> >> is _identical_ to your "idea"? > >> > > >> > Yes. > >> >> > >> >> 3. If so, I cannot see what a meaningful, substantial difference > >> >> between your "irrational idea" and the common "irrational number" > >> >> could be. What - besides pure terminology - is that difference? > >> > > >> > The difference is that an idea has no digits. > >> > >> 2-3. You contradict yourself answering my question 3. that way. As > >> you have agreed to in your answer to question 2. common "irrational > >> numbers" are _identical_ to your "ideas". There are two ways out: > >> > >> 1) Withdraw your answer to question 2. > >> 2) Withdraw your answer to question 3. > >> > >> Which do you prefer? > > > > What is your problem? > > Read 2-3. If you don't understand explain what you did not understand. > > > Common irrational numbers are identical with my "ideas". > > If your "ideas" are _identical_ with common irrational numbers how can > there be "a difference"? > > 1) Either your ideas are _identical_ with irrational numbers then there > is no difference. No difference to the common irratonal numbers. > 2) Or they are different. Different from what you believe is behind these "numbers". > > You have to decide which version of "ideas" shall be valid. Without that > decision you contradict yourself. Common irrational numbers are identical with my "ideas" But the common interpretation of common irrational numbers is false, in particular the assertion of unlimited facility of approimation. Regards, WM |