Cantor Confusion
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From: mueckenh on 17 Dec 2006 07:51 Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Do you believe that today's standards are sound by tomorrow's > > standards? > > Wait a day and we will find out. Hilbert's axiomatization of Euclidean > Geometry done back in 1899 is as good today as it ever was. Aristotelean > logic and semantics have held up very well over 2400 years. Too bad that > Aristotelean physics fell way short. > > The difference between today's math and tomorrows math will be mostly in > scope, not validity. The arithmetic argument that Euclid used to prove > the infinitude of primes He proved that there are more primes than given. > is as valid today as it was 2200 years ago. The difference between today's math and tomorrows math will be that the physical and physiological foundations will be explored and taken into account. This will go the same way as quantum mechanics which has taken into account the observer. There is no trajectory of the electron in reality, and there is no actual infinity in mathematics. Regards, WM
From: mueckenh on 17 Dec 2006 07:54 Dik T. Winter schrieb: > In article <1166167313.573914.50760(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > cbrown(a)cbrownsystems.com schrieb: > ... > > > Unfortunately, following this link led me to a page saying "Das Buch > > > ist nicht in unserer Datenbank gespeichert", which I don't understand, > > > but guess means "That book is not in our database, sadly". > > > > Unfortunately this link was truncated. I try to post it again. > > I do not think that link makes much sense for somebody who does not > understand the German sentence above. That is true. Nevertheless it is available for those who understand German. As it seems that the URL always appears truncated in Google, I give a shorter URL which supplies the 10th chapter http://www.fh-augsburg.de/~mueckenh/MR/P3 Kap 10.pdf or the index http://www.fh-augsburg.de/~mueckenh/MR/P3 Verzeichnis.pdf of the book as a sample. > However, I do understand why > you post that link. How much did publication cost you? My estimate > is about EUR 1000. Far to low a guess! It took me far more than half a year of hard work, which sums up to about 40,000 euro. But I did not write the book in order to become a rich man. I never had that attitude as you can see from my decision to study physics and mathematics. The book is suitable for all students interested in the study of the infinite. It may help some of them to avoid the wrong choice with regard to mathematics. Regards, WM
From: mueckenh on 17 Dec 2006 08:05 cbrown(a)cbrownsystems.com schrieb: > > Consider a binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1 as binary strings. > > So the paths are not "those which can be obtained by repeated > multiplication by 2"; it was /given/ by you at the beginning of your > argument that a path exists for each real number in [0,1]. Yes. And obviously the cardinality of the set of these numbers can be obtained by repeated multiplication by 2, namely 2^aleph0. This is a repetition, repeated as often as there are natural numbers. > Note that you have already given that there are as many paths in the > tree as there are real numbers in [0,1]. We are investigating how we > can construct a surjection of the naturals onto the reals. But we do not presuppose that the reals are uncountable. At least we do not presuppose that ZFC is free of contradictions which may result in two valid proofs with different results. > > Your edges are not > being broken into 2 "shares" each; they are being divided into an > infinite number of shares, and that is very different from 2 shares! For this sake we have mathematics, I mean really correct mathematics, not based on dubious infinite sets, which states that 1 + 1/2 + 1/4 + .... = 2. > > The notion "rational relation" is a further development of the notion > > "relation". While a relation connects elements of a domain to elements > > of a range, the rational relation connects shares of the elements of > > the domain to (shares of the) elements of the range. A simple example: > > > > > > A) Consider the relation: > > > > > > 1 ---> a > > 2 ---> b > > > > Is this drawing supposed to represent a bijection {1,2} -> {a,b}? Of course. > > > > > B) If "1" is divided into two parts, 1#1 and 1#2, we can write (A) also > > in the form > > > > > > 1#1 ---> a > > 1#2 ---> a > > 2 ---> b > > > > Is this drawing supposed to represent a bijection {{1#1,1#2},2} -> > {a,b}? Yes, in case we have {1#1,1#2} yielding a whole edge. > Is this an example of a rational relation? > > (C) > > 1#1 -> b > 1#2 -> b > 1#3 -> b > > 2#1 -> b > 2#2 -> a > 2#3 -> b > 2#4 -> a Sure. It is an example of a rational relation. In order to prove surjectivity of f: {numbers} --> {letters}, we have to show that 1#1 U 1#2 U 1#3 U 2#1 U 2#3 = one element of the set of numbers and 2#2 U 2#4 is also one element of the set of numbers. The latter is false unless 2+1 and 2#3 are empty and no furter shares 2#5 etc exist. > Does this (C) prove the surjection {1,2} -> {a,b} "is possible, even if > it cannot be found"? If instead 1#3 -> a, then does it prove it? Could > you explain why? If 1#3 is a share which contains half of 1, and if all the shares of 2 are of equal size, then we have a surjection. > Wouldn't it be simpler to just give a clear /definition/ of what you > mean by your "rational relation", instead of endless examples? > I gave a clear definition at the biginning of this discussion. I did not give it in the language of ZFC because it seems not possible to do so. > > > I am only interested in your "rational relation" argument. > > > > Fine. Why are you interested only in that argument. And why are you > > interested in it at all? > > I am interested in that argument because it has a surface plausibility > of "mathiness" while being obviously wrong. The reason for the differet result is easy to see. In the tree a real number = path must really exist in order to be accepted. In set theory the real numbers do not exist. It is enough to have a sequence of digits which can be distingusihed from any given sequence. The problem is exactly the same as in my discussion with William Hughes. He must insist that in the Equilateral Infinite Triangle 1 12 123 .... the diagonal must be longer than any line, which is obviously contrary to the definition of a diagonal but required to save the real numbers in ZFC. In the tree we require "intuitively" that every path is separarted from another one not other than by an edge. In effect this leads to the proof that there are no irrational numbers. The "rational relation" is only some abracadabra to veil this "intuitive assumption". > I am only interested in that argument, because to pursue your other > arguments regarding |N| = |R| would be a distraction from the argument > in question. Ok. Then let us stay with the rational relation. Regards, WM PS: I saw that my URL das been posted incomplete again. So I will give a shorter address for a sample. But the book is written in German. Chapter 10: http://www.fh-augsburg.de/~mueckenh/MR/P3 Kap 10.pdf Index http://www.fh-augsburg.de/~mueckenh/MR/P3 Verzeichnis.pdf
From: mueckenh on 17 Dec 2006 08:08 Virgil schrieb: > In article <1166184731.253435.242720(a)l12g2000cwl.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1166092755.336596.309060(a)l12g2000cwl.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Virgil schrieb: > > > > > > > > > WM deceives himself over the number of lines versus the number of > > > > > elements in any one line. The number of lines is not finite but the > > > > > number of elements in any one line is finite. > > > > > > > > Each line differs by 1 element from the preceding line. If the number > > > > of lines is actually infinite then the number of differences must be > > > > actually infinite too. > > > > > > So far so good. > > > > > Their sum is an infinite number. If all elements are there, then also > > all sums are there. > > That is an unjustifiable assumption claimimg that ALL sequences > converge to finite values. > > > > > If not all sums are there, not all elements are > > there. > > There are infinite sequences that do not converge. > > And 1 + 2 + 3 + ... is one of them. And 1 + 1 + 1 + ... also is one of them and, in addition, it is an element of the infinite set. Therefore, in an actually infinte set N, there must be an actually infinite number. Regards, WM
From: mueckenh on 17 Dec 2006 08:14
Virgil schrieb: > In article <1166185295.562736.84050(a)j72g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > > > > > No I simply state that the diagonal proof fails in case of the tree. > > > > > > AS the diagonal proof is not allied to trees, so what? > > > > > Its contradiction is. > > Show me! If you believe in a diagonal longer than any line of the matrix and in paths which have no beginning, then I can't show you anything. A minimum of logic is required for any arguing. > > > > > > > > We can simply count the edges. They are countable. So we an count the > > > > biginnings of separated parts of paths (because every edge is a > > > > beginnng of the separated part of a path, notwithstanding which it may > > > > be). > > > > > > > > > Every finite beginning of a set of paths, however long, is the beginning > > > of as many paths as the set of all paths. > > > > > > So there are as many paths following any such beginning as there are > > > paths altogether. > > > > And there are as many beginnings following such a beginning. > > if a beginning is either a node or an edge, there are fewer beginnings > than unending paths from such a beginning in an infinite binary tree. So, most of the paths have no beginning? Or where, do you think, they begin as separated entities? > > > > > > > Therefore the beginnings of separated parts of the paths are > > > > countable. > > > > > > > > > It is the endings that are not countable. > > > > It is the endings, which do not exist. > > Each path has a head node and a tail consisting of everything else. > > The number of heads in an infinite binary tree is countable, the number > of endless tails from any node is equal to the number of endless binary > strings which is not countable. So how many tails has one head? And where begin these mysterious uncountably many tails? For you it is not a problem that there are more tails than heads and simultaneously that every head has one and only one tail? > > > > No. One can start from some observations like 1+1 = 2. > > That assumes meanings for "1" and "+" and "=" and "2". They can be defined in many ways. That are not axioms. Regards, WM |