Cantor Confusion
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From: William Hughes on 17 Nov 2006 19:28 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. No case ii is there is no largest natural number. and the set of natural numbers has infinite size. (i.e.the natural numbers are counted by omega). You wish to distinguish there is no largest natural number there in no infinite number which counts N (i.e. there is no number omega counting the elements of N) from case iii there is no largest natural number. there are a finite number of natural numbers I don't see the distinction (if a set cannot be counted by an infinite number, what else could it be but finite?) but let us use the name case iv for there is no largest natural number there in no infinite number which counts N (i.e. there is no number omega counting the elements of N) (it is really not a subcase of case ii) > > > > 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, > we can also delete it, because most mathematicians will not accept > physical constraints. > > > > 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). > > 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. In particular we can never complete this > set. We can never put it into a list OK. Knock yourself out. Note however that this case is not consistent with assuming the axiom of infinity. The axiom of infinity says that the set N 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. > > Not here. > > > > b: case ii does not make sense. > > > Case iib 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. > > I don't use case iii. However, you still have to assume case ii. > I use the fact that the diagonal (=bijection, > d_nn) cannot be longer than every line, because it consists of what you > have called the line indexes. [Actually the lines consist of column indexes, it is the columns which consist of line indexes. Since both sets consist of exactly the natural numbers it doesn't really matter] The diagonal contains every column index. No line contains every column index. Therefore the diagonal is longer than every line. Recall, we have assumed case ii, so we have assumed no largest natural number. In particular we have no last line. It is trivial to see that if there is no last line then every line is shorter than the diagonal. How can you claim that the diagonal cannot be longer than every line? > For every element of the diagonal we must > have a line index and, hence, a line. This must hold in the extended > version too. No Recall. The extended version involves adding elements (not lines and columns). Adding one element to every column adds a line, adding one element to every line does not add a column. So if we start with the same number of lines and columns we do not end with the same number of lines and columns. > This is my point of departure. If you say that the > diagonal can be longer than every line > then you say that there are > more natural numbers (elements of the bijection d_nn) than natural > numbers (indexes n). No. I say that the diagonal consists of the sequence of all natural numbers. This means that for any natural number n the diagonal is longer than {1,2,3,...,n}. Since every line can be written in the form {1,2,3,...,n} this means that the diagonal is longer than every line. > Then there is no reason to continue this > discussion. Otherwise we have a contradiction. > > Together with this very systematic account of yours describing our > arguing, we arrive at the present, and probably, final state. > > > 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. > > 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. > No. We have an infinite number of lines. By case ii there is no line with infinite index. Thus there is no element of the diagonal with infinite index. So the set of elements of the diaongal d_nn is exaclty the set of natural numbers < omega. > > (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 > > finit
From: Franziska Neugebauer on 18 Nov 2006 05:50 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > Therefore a set of ordinal number omega must exist (it is the first >> > column) >> >> omega "is" not "the first column". What you may write is, that in >> your sketch the first column _represents_ omega. > > That is a matter of taste. You are in error. Set theory under discussion does not deal with "columns". > I is not uncommon to say N = omega. Since N and omega sometimes denote the same entity you may equate N with omega. There is still no "column". > On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. > Omega is the first transfinite number. Omega is the first transfinite _ordinal_ number. > You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. A treatise in which variables ("n") and number symbols ("0", "1", ...) do not refer to sets is not a treatise _on_ set theory but a treatise of _application_ of set theory, if ever. If (!) that application fails to yield the desired result, you cannot put the blame on set theory. > This all shows that omega is a number. Set theory is not really about "numbers" but about sets. "This all" does by no means "show" that omega is a plain-vanilla (i.e. _natural_) number. > It is said to be the number of natural numbers. In contemporary set theory it is said that omega is _the_ _set_ of natural numbers. The number (cardinality) of omega is named aleph_0. So it is said that the number (cardinality) of _the_ _set_ of natural numbers is aleph_0. "The natural numbers" and "the set of natural numbers" are two names for two different things. This linguistically reflects the difference beween An(n e omega -> p(n)) and p(omega). If you deny that convention there is hardly a common ground for debate. > And omega > n for every n e N. N = omega & a < b := a e b An(n e omega -> n < omega) I can't spot any contradiction. >> Of course not. Two sets X and Y are "bijected" by "pairing" the >> *elements* of the sets not by "pairing" the sets itself. > > If there should be infinitely many lines, then their number should be > omega. The "lists" and "matrices" are creatures of your mind. You should explain "how many" lines you intend to bestow upon the "list". > If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d_ii) i e omega "can" or "must" be longer than every (finite) line occupancy. What every clear-thinking person agrees upon is that the cardinality of the sequence (d_ii) i e omega is greater than the cardinality of (the occupancy of) every single line of that "list" or "matrix". This is due to the _fact_ that there are only finitely many occupied memebers in each line-sequence. > then you say that there are more natural numbers (elements > of the bijection d_nn) than natural numbers (indexes n). I don't see your conclusion. > Then there is no reason to continue this discussion. Otherwise we have > a contradiction. If clarifications are not convenient to you stop discussing whenever you want. But don't ask me not to comment your writings now and then. >> > But an explanation: The set of all sets is an impossible set. >> >> This mathematically reads: There is no set of all sets (in some >> axiomatic systems). Shall I conclude from your "definition by >> example", that you define >> >> possible := exists >> impossible := does not exist > > There is no other interpretation possible, I think. Why do you hesitate? >> >> Sets (including bijections) are neither possible nor impossible. >> >> They exist or do not exist. >> > >> > If they exist, then they are possible. If not, then they are >> > impossible. >> >> Shall I interpret this as "possible := exists, impossible := does not >> exist"? So why don't you simply use the established terms "exist" and >> "not exist"? Maliciousness? > > Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no Zitierwissenschaft (quotation(s)/citation(s) science). > We should pay a bit more respect to the creator of set theory: > "An mehreren Stellen meiner Arbeit werden Sie die Ansicht > ausgesprochen finden, da? dies unm?gliche, d. h. in sich > widersprechende Gedankendinge sind, ..." I don't discuss Cantor's remarks. >> Now the part which you have not answered: >> > I would plead to stop this discussion. We have arrived at a clear > result. You may stop replying to my posts at any time without reason. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. > As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. "ends of terms of lines" is your wording not mine. > In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform you of facts. > you could simply say: > > "There are more natural numbers d_nn than natural numbers n." Your wording not mine. > This sentence is as true as the sentence: The diagonal (d_nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have _you_ ever been arguing on a logical basis with sound arguments? ,----[ <455c4019$0$97239$892e7fe2(a)authen.yellow.readfreenews.net> ] | Adding one element (named "x") to every initial segment gives us: | | ISOFTC' := { {1, x}, {1, 2, x}, {1, 2, 3, x}, ... } | | I have no clue what precisely you mean by adding one element to every | line. Do you mean the set | | L' := { 1, 2, 3, ..., x } ? | | > The fact that it is not maintained proves that your asserted | > bijection does ot exist. | | There exists a bijection between ISOTFC' and L' : | | B' := { <{1, x}, x>, <{1, 2, x}, 1>, <{1, 2, 3, x}, 2>, ... } | | proving |ISOTFC'| = |L '|. `---- F. N. -- xyz
From: Franziska Neugebauer on 18 Nov 2006 06:01 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > Therefore a set of ordinal number omega must exist (it is the first >> > column) >> >> omega "is" not "the first column". What you may write is, that in >> your sketch the first column _represents_ omega. > > That is a matter of taste. You are in error. Set theory under discussion does not deal with "columns". > I is not uncommon to say N = omega. Since N and omega sometimes denote the same entity you may equate N with omega. There is still no "column". > On the other hand omega represents what before Cantor was commonly > abbreviated by oo. Irrelevant to contemporary set theory. > Omega is the first transfinite number. Omega is the first transfinite _ordinal_ number. > You need not interpret n as a set (though you can do it). In contemporary set theory almost everything is a set. A treatise in which variables ("n") and number symbols ("0", "1", ...) do not refer to sets is not a treatise _on_ set theory but a treatise of _application_ of set theory, if ever. If (!) that application fails to yield the desired result, you cannot put the blame on set theory. > This all shows that omega is a number. Set theory is not really about "numbers" but about sets. "This all" does by no means "show" that omega is a plain-vanilla (i.e. _natural_) number. > It is said to be the number of natural numbers. In contemporary set theory it is said that omega is _the_ _set_ of natural numbers [as Virgil pointed out the _ordered_ set]. The number (cardinality) of omega is named aleph_0. So it is said that the number (cardinality) of _the_ _set_ of natural numbers is aleph_0. "The natural numbers" and "the set of natural numbers" are two names for two different things. This linguistically reflects the difference beween An(n e omega -> p(n)) and p(omega). If you deny that convention there is hardly a common ground for debate. > And omega > n for every n e N. N = omega & a < b := a e b An(n e omega -> n < omega) I can't spot any contradiction. >> Of course not. Two sets X and Y are "bijected" by "pairing" the >> *elements* of the sets not by "pairing" the sets itself. > > If there should be infinitely many lines, then their number should be > omega. The "lists" and "matrices" are creatures of your mind. You should explain "how many" lines you intend to bestow upon the "list". > If you say that the diagonal can or even must be longer than > every line, I never said that a diagonal (d_ii) i e omega "can" or "must" be longer than every (finite) line occupancy. What every clear-thinking person agrees upon is that the cardinality of the sequence (d_ii) i e omega is greater than the cardinality of (the occupancy of) every single line of that "list" or "matrix". This is due to the _fact_ that there are only finitely many occupied memebers in each line-sequence. > then you say that there are more natural numbers (elements > of the bijection d_nn) than natural numbers (indexes n). I don't see your conclusion. > Then there is no reason to continue this discussion. Otherwise we have > a contradiction. If clarifications are not convenient to you stop discussing whenever you want. But don't ask me not to comment your writings now and then. >> > But an explanation: The set of all sets is an impossible set. >> >> This mathematically reads: There is no set of all sets (in some >> axiomatic systems). Shall I conclude from your "definition by >> example", that you define >> >> possible := exists >> impossible := does not exist > > There is no other interpretation possible, I think. Why do you hesitate? >> >> Sets (including bijections) are neither possible nor impossible. >> >> They exist or do not exist. >> > >> > If they exist, then they are possible. If not, then they are >> > impossible. >> >> Shall I interpret this as "possible := exists, impossible := does not >> exist"? So why don't you simply use the established terms "exist" and >> "not exist"? Maliciousness? > > Knowledge of literature. Cantor. As you have been told quite some time before: Mathematics is no Zitierwissenschaft (quotation(s)/citation(s) science). > We should pay a bit more respect to the creator of set theory: > "An mehreren Stellen meiner Arbeit werden Sie die Ansicht > ausgesprochen finden, da? dies unm?gliche, d. h. in sich > widersprechende Gedankendinge sind, ..." I don't discuss Cantor's remarks. >> Now the part which you have not answered: >> > I would plead to stop this discussion. We have arrived at a clear > result. You may stop replying to my posts at any time without reason. > You claim that the diagonal of a matrix can be longer than > every line. This is your wording not mine. > As the diagonal is defined to consist of the ends of terms > of lines, this claim is easy to conradict. "ends of terms of lines" is your wording not mine. > In order to defend your claim that there are infinitely many finite > numbers, In the lack of an effective offense I don't have to defend but to inform you of facts. > you could simply say: > > "There are more natural numbers d_nn than natural numbers n." Your wording not mine. > This sentence is as true as the sentence: The diagonal (d_nn) of a > matrix can be longer than every line n. But it shows that there is no > point in further arguing on a logical basis with sound arguments. Have _you_ ever been arguing on a logical basis with sound arguments? ,----[ <455c4019$0$97239$892e7fe2(a)authen.yellow.readfreenews.net> ] | Adding one element (named "x") to every initial segment gives us: | | ISOFTC' := { {1, x}, {1, 2, x}, {1, 2, 3, x}, ... } | | I have no clue what precisely you mean by adding one element to every | line. Do you mean the set | | L' := { 1, 2, 3, ..., x } ? | | > The fact that it is not maintained proves that your asserted | > bijection does ot exist. | | There exists a bijection between ISOTFC' and L' : | | B' := { <{1, x}, x>, <{1, 2, x}, 1>, <{1, 2, 3, x}, 2>, ... } | | proving |ISOTFC'| = |L '|. `---- F. N. -- xyz
From: mueckenh on 18 Nov 2006 08:25 Virgil schrieb: > 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. I use the "hidden assumption" that there are not more natural numbers d_nn than natural numbers n. If you can accept that the are more natural numbers than natural numbers, then we have different world views which do not fit together and cannot be united. So we should stop here. Your mathematics is not mine and vice versa. I assume that part of my mathematics belongs to absolute truth which does not rely on arbitrary axioms. Perhaps sometimes an alien civilization will be contacted. It would be interesting to get to know their mathematics. But the chances are very low. Anyhow it is not useful that we comment the contributions of each other any longer, because the positions are settled and fixed. =================== > If WM wisheds to claim that there is some n for which the length of the > diagonal is NOT greater than n, let him produce it now, or forever hold > his peace. I claim that there cannot be a diagonal element where lines are lacking. That means: There are exactly as many lines as there are diagonal elements. And there are lines as long as the diagonal is. Because without lines the diagonal cannot exist. This "hidden assumption" leads to the necessity of aninfinite line (number) if the diagonal (the set of numbers) is infinite. This concluison excludes an actually infinite diagonal. Regards, WM
From: mueckenh on 18 Nov 2006 08:27
Lester Zick schrieb: > >Sqrt(2) does exist as the diagonal of the square. But I call that an > >idea in order to distinguish it from numbers which can be written in > >lists and can be subject to a diagonal proof. I call only those > >entities numbers which can be put in trichotomy with each other. > > I don't know what a diagonal proof and trichotomy may be. Excuse me. 1) The diagonal proof by Cantor shows that any list of real numbers is incomplete. For this purpose we use a list (= injective sequence) of real numbers and exchange the n-th digit of the n-th number. The changed digits put together yield a real number which differs from each list entry at least at one position (it differs at position n from the n-th list entry). This proof requires that all digits of numbers like sqrt(2) do exist and can be exchanged. That assumption is wrong. 2) The fact that for each pair of numbers a and b we have a < b or a = b or a > b is called Trichotomy. For such numbers as P = [pi*10^10^100] and Q = (the same number but the last digit exchanged by 5) the order by size cannot be determined. [x] denotes the integer part of x. Regards, WM |