Cantor Confusion
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From: Franziska Neugebauer on 17 Nov 2006 13:50 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. > (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. -- xyz
From: Virgil on 17 Nov 2006 14:23 In article <1163768117.616662.206700(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > In my approach, the lines contain unary representations of natural > > > numbers. > > > > What the lines contain (occupancy) does not effect the number of > > sequence menbers. > > It does, if the members enumerate themselves. This is the case in the > present EIT: > > 1 > 12 > 123 > ... > > >> Equivocation: "Adding one element" names two different things > > >> (changing the occupancy vs. changing the domain). > > It names two different things only if there are two different things in > the initial bijection. But that cannot be the case because these things > are identical (except that one is noted vertically and the other one > horizontally.) Hence, any difference can only be that the nunmber of > these things is different. That is what I proved. I have yet to see any WM "proof" that is mathematically or logically satisfactory. There are always hidden assumptions that are unwarranted. > > > > The question is whether there is a bijection between the > > > initial segments of the first column and the lines like > > > > > > 1 > > > 2 > > > 3 > > > ... > > > n > > > and 1,2,3,...n. > > > > ISOTFC := { {1}, {1, 2}, {1, 2, 3}, ... } > > L := { 1, 2, 3, ... } > > > > B := { <{1}, 1>, <{1, 2}, 2>, <{1, 2, 3}, 3>, ... } > > > > The bijection between the initial segments of the first column (ISOTFC) > > and the lines (L) is explicitly contructed. I have named it "B". I > > cannot see any reason to debate about the existence of B. > > This bijection contains only finite sets. If you add one element to a > finite set, then you get a finite set. If you add an element to an > infinite set you get a set of ordinal number omega + 1. If you add one element to an infinite set,, you do not necessarily get an ordered set at all. If you add one element to an infinite sequence, to follow all the members of that sequence, you no longer have a proper infinite sequence. > > Therefore a set of ordinal number omega must exist (it is the first > column) but it does not appear in your bijection. The set of pairs in that bijection itself forms a set of order type omega.
From: Virgil on 17 Nov 2006 14:28 In article <1163768231.302005.176200(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > > The natural numbers count themselves. Bijection of initial segments of > > > column and lines > > > > > > 1 > > > 2 > > > 3 > > > ... > > > n <--> 1,2,3,...n > > > > A n e omega. > > > > > If there is no infinite number then there are not infinitely many > > > numbers. > > > > 1. If *where* is no infinite number? > > There is no infinite number among the lines 1,2,3,...,n. There can be > no bijection with the completed first column. For having only lines 1,2,3,...,n, the "completed" columns are all no longer than n. > > > 2. Define "infinitely many" [, please.] > > Here I mean with "infinitely many" a number which, like the column, by > adding 1 element yields a transfinite number omega + 1. That is not a valid definition of infinitely many. At least not valid in any mathematical sense. What WM seems to think of as valid or invalid seems to have little or nothing to do with either logic or mathematics.
From: Lester Zick on 17 Nov 2006 14:36 On 16 Nov 2006 10:36:55 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On Thu, 16 Nov 2006 01:35:12 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >Virgil wrote: >> >> In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > It is difficult to answer this question, because the expression "set" >> >> > is occupied in modern mathematics by collections of elements which are >> >> > actually there (you don't know what that means, imagine just a set as >> >> > you know it). Such infinite sets do not exist. >> >> >> >> While infinite collections in any physical sense are not possible, why >> >> are imaginary infinities, such as sets of numbers must be, unimaginable? >> >> Why are square circles unimaginable? > >They aren't. We can define "the set of squares S". We can >define "the set of circles C". Most of us (except you, >apparently) can easily imagine defining the set >SC = C intersect S. Ah yes. The old "define different elements one-at-a-time technique" then talk about their intersect. So do tell us, Randy, exactly where is this interesect you and other modern zen mathematikers have no problem "imagining" but I do? Disneyland? Through the looking glass? >It happens under the rules of Euclidean geometry that >the set SC is empty. But that doesn't stop most of us (except >you, apparently) from imagining a geometry where it is not >empty. Of course not. That's why we keep you around. For entertainment. No wonder zen mathematikers don't proclaim the truth of what they say. They use Euclidean geometry to draft definitions for squares and circles then proclaim that they'd rather do non Euclidean geometry instead and it's honky dorry not to be too finicky about the truth of their definitions because truth is irrelevant to zen mathematics in any event and definitions can't be true or false. >Furthermore, the fact that SC is empty in Euclidean geometry >would be a theorem. There's nothing inherently wrong >with defining the set. I just did. It's called square circles. Only I just didn't bother to go through all the pretentious malarky you find so appealing in modern zen math to conceal what you're doing from those who still believe in the truth fairy. I just rhetorically suggested they're unimaginable. You know, just like truth is unimaginable in modern math? > Most of us (except you, apparently) can >see that forming such a definition tells us nothing about >whether the set is empty or not. It's just a shorthand, the name >of the membership test for SC. Of course not. No one ever accused modern zen mathematikers of being able to tell anything about what they're doing. Least of all moi. And certainly not when they're doing non Euclidean geometry with Euclidean definitions which can't be true or false by definition since truth is an alien concept and completely irrelevant to modern zen mathematics. >Most of us (except you, apparently) can then easily imagine >a proof that SC is empty which relies on argument from >axioms, not such handwaving as "the definition is false". "The definition is self contradictory" is handwaving? Now I can readily imagine "self contradiction is false" to be handwaving but not "the definition is self contradictory". That's either true or not. Oops I'm sorry. There's that pesky t-word again mathematikers hate. Just how do you work around the t-word exactly? I mean mathematikers have no trouble at all appealing to the f-word. Or at least one of them. I know. Let's evaluate the t-word of the following proposition: "There is a real number line." So how would the truth fairy analyze this? "Yes"? "No"? "Maybe"? "I don't know"? "This is not a mathematical proposition"? "The t-word is not relevant to modern math but the real number line is"? Or perhaps "The intersect of the set of real number line with the set of t-word is empty most of the time, some of the time, or all the time we're not doing non Euclidean geometry"? ~v~~
From: Virgil on 17 Nov 2006 14:37
In article <455de015(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > William Hughes wrote: > > 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. > > > > 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. > > > > iii: There are a finite number of natural numbers, but > > there is no largest natural number. > > > > > > Case i is highly counterintuitive because of "What > > about (n_L+1)?". > > > > I claim case 2 (existence of N and no largest number by the > > axiom of infinity, infinitely > > many numbers the fact that all the natural numbers have been used > > to provide sizes of other sets). > > > > You claim case iii (finite number based on an argument that only the > > integers that > > will actually be named can be said to exist, no largest number based on > > the > > fact that the largest number is unknowable before the end of the > > universe). > > > > If case ii is correct then there can be infinitely many lines without > > any line that does not correspond to a natural number. In this > > case the bijection exists. > > > > You want to show two things. > > > > a: case iii makes sense, i.e. there can be a finite number > > of natural numbers but no largest natural number. > > > > b: case ii does not make sense. > > > > > > To do a you need to define finite and show that this > > definition does not lead the existence of a largest natural. > > (note that an argument that there are only a finite number > > of natural numbers because there are only a finite number > > of available bits does not help. You still need to define finite). > > > > To do b you need to assume case ii and show this leads > > to a contradiction. However, when you do this you cannot > > use case iii. In particular, we know that asumming case ii > > means that we can find a bijection between a matrix with > > an infinite number of lines, but for which every line is finite. > > (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) > > > > - William Hughes > > > > Hi William - nice summary. > > The difference boils down to potential (countable) infinity vs. actual > (uncountable) infinity. The set of naturals is countable, and therefore > only potentially infinite. If we define an infinite set as one where > expressing the indexes of all elements in a finite base digital system > requires unending strings for most elements, then only uncountable > infinity fits that bill. Then, according to TO, potentially infinite is a form of infiniteness, rather than, as he was wont to say , a form of finiteness. Perhaps TO is capable of learning things despite all the evidence to the contrary previously presented here by him. |