Cantor Confusion
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From: Lester Zick on 17 Nov 2006 16:51 On 17 Nov 2006 12:46:08 -0800, "David R Tribble" <david(a)tribble.com> wrote: >Dik T. Winter wrote: >>> If you want to find absolute truth you should not look at mathematics. >> > >david petry schrieb: >>> Perhaps we should replace "absolute truth" with "culturally neutral >>> truth", or in other words, truth without any cultural, religious, or >>> philosophical bias. [...] Thinking about this question >>> leads most of us to believe that there is a core of mathematics which >>> every such civilization will accept. >> > >mueckenh wrote: >> without axioms, yes. For instance: I + I = II (after translating "+" >> and "="). Therefore I call this an absolute truth. > >Which axioms are you using to describe the "+" and "=" operators? None. ~v~~
From: David R Tribble on 17 Nov 2006 17:49 david petry schrieb: >> Thinking about this question leads most of us to believe that there >> is a core of mathematics which every such civilization will accept. > mueckenh wrote: >> without axioms, yes. For instance: I + I = II (after translating "+" >> and "="). Therefore I call this an absolute truth. > David R Tribble wrote: >> Which axioms are you using to describe the "+" and "=" operators? > Lester Zick wrote: > None. Then how can you know it is true? In fact, it looks wrong to me. I think it should be I + I = I. But perhaps I have a different implicit understanding of "+" and "=". What do your "+" and "=" mean?
From: Virgil on 17 Nov 2006 18:04 In article <455e0473$0$97237$892e7fe2(a)authen.yellow.readfreenews.net>, Franziska Neugebauer <Franziska-Neugebauer(a)neugeb.dnsalias.net> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> > >> http://mathworld.wolfram.com/InitialSegment.html > >> > >> What you call "complete initial segment" is not an *initial* segment > >> but the whole set of lines. > > > > What is a name? > > You should rephrase what you mean. > > > What *counts* is this: It is asserted that the number of natural > > numbers is omega > > "It" is asserted that the *set* of natural numbers is (does exist) and > is named omega. The *cardinality* of omega is aleph_0. Actually it is asserted that the ORDERED set of naturals is named omega. Absent order one cannot have omega, and sets need not be considered as ordered sets, even when there is an obvious order one can use. > > > (the elements of the first column = number of lines). > > The ordinal number of lines is omega. Its cardinal number is aleph_0. > Neither omega nor aleph_0 are /natural/ numbers. Neither omega nor > aleph_0 are elements of omega. Period. > > > By adding 1 element to the first column we see that, if this assertion > > is correct, the number increases to omega + 1. > > It depends on how you in the present case define "adding 1 element to > the first column". As pointed out: > > L' := { 0, 1, 2, ..., x } is of ordinal type omega + 1 > L'' := { x, 0, 1, 2, ... } is ("still") of ordinal type omega > > Besides of this | L' | = | L'' | = aleph_0 is valid. > > What "L + 1" do you mean? L' or L''? > > > But are there enough lines? > > There are as many lines as you have defined. > > > No. > > How did you hit on that? > > > That means: The finite natural numbers are not sufficient > > to stand in bijection with all omega elements of the column, > > Here you are: > > B := { <0, 1>, <1, 2>, <2, 3>, ... } > > B is an explicit bijection between the naturals (elements of omega) and > the numbers in the first column. > > > but only with the finite ones. > > Which L are you talking about? L, L' or L''? > > > Therefore the unavoidable conclusion is: There are not omega finite > > numbers. > > Please come to a decision what case (L, L' or L'') you are talking > about. > > F. N.
From: Virgil on 17 Nov 2006 18:28 In article <1163798651.845452.180040(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > The natural numbers count themselves. Bijection of initial segments of > > > column and lines > > > > > > 1 > > > 2 > > > 3 > > > ... > > > n <--> 1,2,3,...n > > > > > > If there is no infinite number then there are not infinitely many > > > numbers. > > > > This is clearly the point of contention. > > > > Consider, N, the set of all natural numbers. > > By definition N only contains natural numbers. > > > > Cases > > i: There is a largest natural number,n_L, then N={1,2,3,...,n_L} > > In this case there are n_L natural numbers. > > Deleted. > > > > ii: There is no largest natural number. We will > > write this as N={1,2,3,...} (the ... represent only natural > > numbers). Set N has infinitely many > > elements. > > Please distinguish: > iia: There is no number counting the elements of N. > iib: There is a number omega counting the elements of N. There is an ordinal number "counting" the elements of the ordered set N. There is a cardinal number counting the elements to the unordered set N. There is no natural number counting the members of N whether considered as being ordered or not. > > > > iii: There are a finite number of natural numbers, but > > there is no largest natural number. > > > Deleted. Athough this assertion is the only correct one, in my opinion, Which shows how little WM's opinion is worth. > we can also delete it, because most mathematicians will not accept > physical constraints. Why accept physical constraints in non-physical situations? > > I claim case iia: There is a potentially infinite sequence N = > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > treat all of its elements. Speak for yourself only. What you cannot do does not necessarily limit others. > In particular we can never complete this > set. We can never put it into a list. We don't have to, it is already a list. > I use the fact that the diagonal (=bijection, > d_nn) cannot be longer than every line Maybe not in WMworld, but in any mathematical world, that particular diagonal is at least n+1 places long for ever n in N, which makes it longer that line n for every n in N. Since every line is a line n for some n in N that proves : The diagonal is longer than every line. To refute this WM must produce some (finite) line that is at least as long as the (infinite) diagonal. Which WM has not done and cannot do. > This is exatly the point where you have to admit that there are more, > namely omega, natural numbers (d_nn), than natural numbers n < omega. Why admit falsehoods? We do not need any such thing until WM produces a line as long as the diagonal. > > > (There is a line for every natural number, but no other > > line. Therefore by ii there are an infinite number of lines. > > Line n will be {1,2,3,...,n} and have length n, which is > > finite) > > You have to explain the difference appearing after extending every line > and every initial segment of the column. You cannot do it by claiming > different things, because for all n, the initial segments and the lines > are identical (see above). Every line in this model is of length n for some finite natural n in N. For every line, of length some finite natural n, there is a position n+1 in the diagonal making the diagonal longer than that line. WM claims that there is a line at least as long as that diagonal but cannot produce any evidence that any such line ever existed.
From: Virgil on 17 Nov 2006 18:40
In article <1163798879.429134.252490(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > "More than any finite number" is already the number of elements of the > first column. But it is said that a set of finite numbers suffices to > build this infinite set. This is wrong but it is not easy to show the > error. It is, in fact, impossible to show an error here, because the error lies only in the eye of one beholder, and most of our eyes do not lie that way. > Therefore I add one element in order to have a set which has not > only infinitely many elements but which provably must have also a > transfinite element w+1. Adding one element to an ordered set need not be done at any particular point in the ordering, and if one does it at random, there is effectively no chance of it being inserted where WM wants to have it inserted. It makes much more sense to use the Hilbert Hotel scheme for inserting new elements, by putting any new one at the front and shifting all the others down one place. > > In order to explain it, one must assume that the diagonal of the > equilateral infinite triangle (IET) must be longer than every line. But > the diagonal consists of ends of lines and nothing else. Therefore this > assertion is a self-contradiction: The set of ends of lines (d_nn) must > contain more elements than the set of lines n. Deliberate misrepresentation. To say that the diagonal is longer that every line only means that for every line, of length n, there is an end of line d_{n+1}{n+1} beyond it. Which is obviously the case, though WM scream and holler ever so loudly in opposition. > > Regards, WM |