An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 10 Aug 2006 17:36 In article <1155242105.069297.90260(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > Dik T. Winter schrieb: > > >> The last line of the triangle will have equal width as the length of the > > >> triangle. If we get at an infinitely long triangle, there is no last > > >> line. > > > > > > > Mueckenh wrote: > > > And there is no definite number aleph_0 of lines. > > > > > > But if there were actually infinitely many, namely aleph_0 lines, then > > > already the first 10 % of lines were infinitely many. And 90 % of the > > > lines were infinitely long. > > > > What is 10% of Aleph_0? If you start counting your lines, at what > > point do you know that you've counted the first 10% of them? > > At what point do you know that you have all aleph_0 lines? When you have them all.
From: David R Tribble on 10 Aug 2006 20:07 Mueckenh wrote: >> And there is no definite number aleph_0 of lines. >> >> But if there were actually infinitely many, namely aleph_0 lines, then >> already the first 10 % of lines were infinitely many. And 90 % of the >> lines were infinitely long. > David R Tribble schrieb: >> What is 10% of Aleph_0? If you start counting your lines, at what >> point do you know that you've counted the first 10% of them? > mueckenh wrote: > At what point do you know that you have all aleph_0 lines? When your list (set, sequence, list) has a line for every natural. When you can show that every line is denumerated by a natural, then you know you've got Aleph_0 lines. So tell me, where in this list is the 10% point?
From: Dik T. Winter on 10 Aug 2006 22:27 In article <1155210016.182960.266320(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Example: > > > The third 1 of 0.111... can index the third 1 of 3 = 0.111 as well as > > > the third one of 5 = 0.11111. > > > > It is a strange formulation to state that the third 1 can index the third 1 > > of something else, because it is *not* the 1 that indexes, it is its > > position number. I would rather formulate it as: the index number of the > > third 1 can index the third 1 of something else. > > That is the correct formulation. It clearly shows the symmetry of > indexing. > > > > > General: Instead of third, 3, and 5 you can choose n-th, n, and m > n. > > > This shows the symmetry of the relation "indexing". > > > > But there is no symmetry as you appear to assume. > > You correct formulation above makes the symmetry obvious. > > Further: Indexing the digit number n is equivalent to covering the > string up to digit number n. A finite string can never cover an > infinite string. But *that* is irrelevant. Consider K = 0.111... . For each n we can index digit number n of K and so cover K up to digit number n. > Even a finite string of an infinite set of finite > strings can never cover an infinite string. What is the relevance? I never stated that K can be "covered" by some n. I rather state that K can not be covered by any n, just as you state here. This does *not* mean that there are digits of K that can not be indexed by some n, nor that the digits upto digit position n of K can not be covered by n. I fail to see what you are arguing here. > > > So infinitely is not more than finitely? > > > > How do you conclude that from what I write? > > From your assumption that infinitely many digits can be indexed and > covered by finitely many digits. This is rubbish. Given z = 0.1111, by the digits I can only index the first position of any number. It is by the *digit positions* that I can index the first four digits of any number. My assumption is *not* that infinitely many digits can be indexed and covered by finitely many digits; it is that infinitely many digits can be indexed and covered by infinitely many digit positions. > Don't forget: The infinite set of natural numbers contans only natural > numbers, i.e., finite strings of 1's in unary representation. Yes, but there is no bound on the index positions, and hence the number of index positions is infinite. > > > According to that the number of 1's of 0.111... is aleph_0 which is > > > larger than the number of 1's of any finite number. This forbids > > > complete indexibility. > > > > Why is that forbidden? The number of natural numbers is also aleph-0, > > so it appears to me there is a perfect match possible. nevertheless, > > you state it is impossible. By the axiom of infinity, the set of > > naturals does exist, and by Cantor, the cardinality is aleph-0. Also, > > that set does not have a last element. > > But there is no natural number aleph_0. Where do I state that? > Why can't you learn that the > asserted infinity of the number of numbers is completely irrelevant. That is just opinion. > What counts (in the true sense of the word) is how many digits a number > has. And every natural number has only finitely many digits, regardless > of how many numbers there are. I never contradicted that. Why are you always saying that I claim something which I do not claim? > > > > Not if you need *all* list numbers to index > > > > > > You say "not". That "not" is understandable but nevertheless it is > > > wrong. Even if all list numbers are needed, none of them is infinite > > > (by definition each one is finite; that is why there is no infinite > > > set of finite numbers). > > > > Yes, you keep stating that, but the axiom of infinity asserts that there > > is. > > Please stop your current intermingling of infinity in sizes of numbers > and number of numbers. > There is no ifinite size. from hat I conclude that there is no infinite > set. Please stop claiming you found an inconsistency with the axiom of infinity when your basic assumption already is in contradiction with it. > > But if you claim that the set if finite numbers is finite, there is > > a largest number. > > There is no largest number. The set of of natural numbers is infinite, Eh? Above you wrote that there is *not* an infinite set. Contradicting yourself? > but they don't exist all together. If they did, then there were > infinite numbers, as you always seem to believe. Pray give a mathematical definition of "exist". > > What is the successor of that largest number? I would > > state that successor is also finite. A contradiction, so the set of > > finite numbers is not finite. > > There is no infinite natural number. Because there is no infinite > finite number - and every natural number is finite. Even the axiom of > infinity does not state the contrary. (But if the axiom of infinity had > to be satisfied, then a finite infinite number would be required. > Compare the staircase.) Again an assertion, without proof. > > > Absolutely uninteresting how many there are. Each one is finite. > > > > Yes, and so what? There are nevertheless infinitely many of them. > > Can't you understand that it is uninteresting how many there are? Can't you understand that in mathematics it is on occasion very interesting to compare infinite sets? > Indexing is not executed by a many of numbers but each time by only one > number. And this one number is always finite - each time! > > THERE IS NO INFINITE STRING OF 1's IN THE UNARY REPRESENTATION OF > NATURAL NUMBERS. And indeed, I have said this over and over again. So what is the argument? > > > Indexing all positions of 0.111... means covering all positions, i.e., > > > covering the whole number 0.111... . If this were possible, then the > > > list would contain at least one number 0.111... with at least aleph_0 > > > positions filled with 1's. That, however, is impossible. > > > > Not so. For each n a finite segment of 0.111... is covered, but there is > > no n that covers 0.111... completely. > > Therefore 0.111.
From: Dik T. Winter on 10 Aug 2006 22:30 In article <1155242105.069297.90260(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > David R Tribble schrieb: .... > > What is 10% of Aleph_0? If you start counting your lines, at what > > point do you know that you've counted the first 10% of them? > > At what point do you know that you have all aleph_0 lines? When you start at 1, at no point. But the axiom of infinity asserts that the set of all aleph-0 lines does exist. I think you are asserting a new axiom: 10% of the lines does exist. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 11 Aug 2006 12:36
Virgil schrieb: > In article <1155067333.259873.193700(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > The v. Neumann model of natural numbers is just the model of segments. > > n = {0, 1, 2, ..., n-1} > > Every natural number is finite. > > Hence every union of finite segments is a finite segment. > > Not necessarily. You assume that every union of finite segments is a > finite union, which begs the question. which is obviously true if the v. Neumann model is assumed, because every union of sequences is a natural number and every natural number is finite. Do you believe in infinite natural numbers? Or is it true that you never really thought about that topic, i.e., how infinitely many finite numbers can exist without one or the other being infinite? Regards, WM |