Cantor Confusion
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From: Andy Smith on 27 Jan 2007 17:39 Andy Smith <Andy(a)phoenixsystems.co.uk> writes > >That doesn't mean that the reals are countable - far from it. On the >hypothesis that the rals are countable, we ahave a situation where, on >our list, for all entries n in the hypothetical list n e N, the number of >zeroes between the last non zero term and the diagonal bit increases >monotonically as log(n)-loglog(n). So there can never be any non-zero >bits crossing the diagonal. But what about e.g. sqrt(2) ? Any real in the >set with an infinite binary expansion will intersect the diagonal. > Should have said of course, careless,"increases monotonically as n-log(n)": -- Andy Smith
From: Carsten Schultz on 27 Jan 2007 18:10 mueckenh(a)rz.fh-augsburg.de schrieb: > > I do not want to prove something about a maximum of the set N. I am > interested in the fact that every set of natural numbers has a finite > maximum. > Everyone convinced now that every attempt at argumentation is futile? -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page.
From: G. Frege on 27 Jan 2007 18:47 On Sat, 27 Jan 2007 22:26:52 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > I think that Cantor's diagonalisation argument has a terminal flaw. > You have earned crank status now. Congratulations! F. -- E-mail: info<at>simple-line<dot>de
From: Dave Seaman on 27 Jan 2007 19:10 On Sat, 27 Jan 2007 22:26:52 GMT, Andy Smith wrote: > Randy Poe <poespam-trap(a)yahoo.com> writes >> >>I do know that the set of infinite binary strings is uncountable, >>but I know that because it is easily proven. >> > Well, at the high risk of ridicule and a rather lower probability of > assassination by irate set theorists, I think that Cantor's > diagonalisation argument has a terminal flaw. > The flaw is that Cantor's argument, disregarding any qualms about the > infinite sets involved, just demonstrates that one cannot construct an > infinite list of all permutations of binary strings. this is not the > same as showing that you cannot construct an infinite list of all reals, > because, as has been pointed out, reals may have more than one > representation in any base. The size of the set of real numbers does not depend on what base is used to represent them. Therefore, we are perfectly free to choose base 10, rather than base 2. We can arrange the proof so that all the digits of the generated number are either 4 or 5, so that the number thus generated does not have a dual representation and therefore your objection does not arise. There is another way to see that the set of binary strings can be injected into the reals. Given a binary string, we can replace each "1" by a "2" and interpret the resulting string in base 3. As a result, no to strings represent the same real number. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Dik T. Winter on 27 Jan 2007 20:36
In article <1169822145.767270.204980(a)v45g2000cwv.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 26 Jan., 13:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1169804998.026115.299...(a)q2g2000cwa.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > On 25 Jan., 17:03, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > No. An element of a set S is written as x in S = {x| ...}. > > > > Yes, if there is a single x. > > I defined a sequence as an element of the "set of corresponding > sequences". Therefore there are single elements, as usual, written in > front of the "|". The k-th element is a_1k, a_2k, a_3k, ... You are able to invent new notation without knowing it. Your notation is ambiguous, to say the least. You wrote: > S = { a_1k, a_2k, a_3k, ... | a in {0, 1}, k in N } without explicit notational clarification it is not even clear what this does mean. I already proposed a much clearer notation: S = {[a_1k, a_2k, a_3k, ...]| a in {0, 1}, k in N } where S consistes of the sequences: [a_1k, a_2k, a_3k, ...]. If you want to present a sequence as a single unit, present it as a single unit. In the notation: S = { x_k, y_k | k in N } I do *not* see a set of pairs, bit a set of elements x_k and y_k. When it is a set of pairs, the proper notation is: S = {(x_k, y_k)| k in N } > >You can also write: > > S = { x_i, y_i | i in N } > > meaning S contains as elements all x_i and y_i for i in N. Your notation > > is a novelty invented by you. > > New ideas often require new notations. In that case you have to *explain* your notation. But your idea could easily be formatted within standard notation, not new notation needed. And new notations should *never* be ambiguous. > > > > > However, if L is a set of > > > > > limits L_k > > > > > L = { L_k | k in N } > > > > > and S is the set of corresponding sequences > > > > > S = { a_1k, a_2k, a_3k, ... | a in {0, 1}, k in N } > > > > > then L cannot be larger than S, because there cannot be more limits > > > > > than sequences. > > > > this makes no sense if S is *not* a set of sequences. > > > > > > S is a set of sequences. > > > > You say so, inventing completely new notation. > > Sets of sequences are not so new. Indeed, but your notation is new. > > > > Consider the > > > > following: > > > > L = { L_k | k in {0, 1, 2}} > > > > S = { a_1k, a_2k, a_3k, ... | a in {0, 1}, k in {0, 1, 2} } > > > > If S is a set the cardinality of S is at most 2. > > > > > > Cardinality is |S| = 3. Its elements are the sequences number k = 0, 1 > > > , and 2. > > > > Wrong. > > Oh, I thought you had understood? A set of three sequences has > cardinality 3. Yes, but in your notation of the set you do not use the proper notation for a sequence as element of a set. > > > > If S is an ordered > > > > *multi-set*, the ordinality is omega. The cardinality of L is 3. > > > > S is certainly not an ordered set because it contains multiple > > > > identical elements. > > > > > > S is ordered by the numbers k. > > > > Wrong. > > So the sequences S_0, S_1, S_2 are unordered? Pray consider the notation S = { [ a_1k, a_2k, ... ] | a in {0, 1}, k in N } or somesuch. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |