Cantor Confusion
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From: David Marcus on 19 Nov 2006 16:12 Virgil wrote: > In article <1163938774.893086.200910(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > For example, it is definitely the case that for any given position n, > > > the nth digit of sqrt(2) is, at least in principle, determinable, > > > however impractical it might be to carry out such a determination. > > > > No, it is clear for everyone who is not a fanatic that in principle and > > in praxis not every digit of sqrt(2) is determinable. > > I did not say "every", I said "any". Does WM claim that there is ANY > digit in the decimal expansion of sqrt(2) that is not, at least > theoretically, determinable? I'll bet he says that you can determine any digit, but not more than 10^ (10^100) of them at a time. > If so he is a fool. -- David Marcus
From: Virgil on 19 Nov 2006 16:18 In article <1163939685.919928.156930(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Virgil wrote: > > > > > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > >> Virgil schrieb: > > [...] > > >> And there are lines as long as the diagonal is. > > > > > > Name one. > > The elements of the diagonal are a subset of the line ends. The SET of elements of the diagonal equals the set of all line ends, but that does not name any line as being as long as the diagonal. It is easy to see that for every line in WM's list the diagaonal must contain at least one more character than that line.
From: Virgil on 19 Nov 2006 16:36 In article <1163940320.577431.95910(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > > > > > > > > > 1 > > > > > > > 2 > > > > > > > 3 > > > > > > > ... > > > > > > > n <--> 1,2,3,...n > > > > > > > > > > > > Please distinguish: > > > > > iia: There is no number counting the elements of N. > > > > > iib: There is a number omega counting the elements of N. > > > > > > > No. You are trying to show that assuming case iia leads > > to a contradiction. > > No. Case iia does not lead to a contradiction. That depends on (1) the axiom system in which one is operating and (2) the definition of "number" one is using. > > > To do this you need to assume case iia. In > > particular > > you are assuming > > > > - the set, N, of all natural numbers exists > > - the set N is infinite. > > - N has no last element > > Yes. > > > > > The diagonal contains all the d_nn. No line contains all the d_nn. > > Therefore the diagonal is longer than every line. > > The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! > > > > > The set of all lines contains every d_nn. No single line > > contains every d_nn. The diagonal contains all d_nn. > > The diagonal is longer than every line. > > The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! > > > > > > > > So we have two results: > > > 1) The diagonal must be longer than every line. > > > 2) The diagonal cannot be longer than every line. > > > > No. The diagonal contains all the d_nn. No line contains > > all the d_nn. The diagonal is longer than every line. > > The diagonal consists of line indexes, i.e., of the line ends. > Therefore it is a subset of the line indexes. But no single line contains all the line indices, so that is totally irrelevant when comparing single lines to the diagonal. No matter how you squirm, WM, you are wrong here! > > > > > > > > Only if the complete column does exist. But just that is wrong. > > > > > > > If we assume case iia the complete column does exist.. > > The assumption leads to a contradiction. Therefore the assumption is > false. Or some other assumption necessary for existence of the alleged contradiction arises is false. Very few assumptions are sufficient in isolation to produce contradictions, they usually have to contradict something else. In which case, it can be any of the several assumptions leading to that contradiction which one may wish to reject. > > > > No. it just shows that the assertion of an actually infinite set of > > finite > > numbers leads to results that you do not like. However, these > > are not contradictory results. > > These are: > 1) The diagonal must be longer than every line. > 2) The diagonal cannot be longer than every line. (2) is an unprovable assumption. WM's attempt to demonstrate it are all trivially invalid. > That is in my opinion a contradiction. The contradiction is not in the axiom but in WM's assumptinos over and above the axioms.. > > > > Why does addition of one element yield different results for columns, > > > diagonal and lines? > > > > The columns and the diagonal both contain an infinite initial > > segment. No line contains an infinite initial segment. > > I know that. But you should recognize that this is a contradiction. As all that is contradicted are WM's beliefs, that is WM's problem, not ours. > If > not, try to transpose the matrix. Easy enough, just list each line vertically downwards and align the lines horizontally with first members in a row instead of a column. > > The assumption has been assumed. The assumption leads to a > contradiction. Therefore the assumption is false. AS it is only WM's assumption that is false, everyone else can be happy. > > Regards, WM
From: Virgil on 19 Nov 2006 16:53 In article <MPG.1fca92ea5aac84e3989962(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <1163938774.893086.200910(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > For example, it is definitely the case that for any given position n, > > > > the nth digit of sqrt(2) is, at least in principle, determinable, > > > > however impractical it might be to carry out such a determination. > > > > > > No, it is clear for everyone who is not a fanatic that in principle and > > > in praxis not every digit of sqrt(2) is determinable. > > > > I did not say "every", I said "any". Does WM claim that there is ANY > > digit in the decimal expansion of sqrt(2) that is not, at least > > theoretically, determinable? > > I'll bet he says that you can determine any digit, but not more than 10^ > (10^100) of them at a time. > And for the diagonal, you only need one at a time. > > If so he is a fool.
From: Virgil on 19 Nov 2006 16:59
In article <45605cf3$0$97253$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: > >> Virgil wrote: > >> > >> > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, > >> > mueckenh(a)rz.fh-augsburg.de wrote: > >> > > >> >> Virgil schrieb: > >> [...] > >> >> And there are lines as long as the diagonal is. > >> > > >> > Name one. > > > > The elements of the diagonal are a subset of the line ends. > > Though Virgil posed this question: Name one single _line_ which is as > long as the diagonal ist. > > F. N. It is a question that WM dare not face directly, as facing it would show up his errors unmistakably. |