Cantor Confusion
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From: mueckenh on 20 Nov 2006 08:58 Franziska Neugebauer schrieb: > 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. Name one single element of the diagonal which is not contained in a line (which contains this and all preceding elements). This is what I call the one-eyedness of set theory: Its proponents see that for every line, there is a diagonal element not contained in this and all preceding lines. But they don't see, or at least dispel it, that there is no element of the diagonal which is outside of any line. The first observation leads to the theorem: The diagonal is superset of all lines. The second observation (together with the fact that every line is a superset of all preceding lines) leads to the theorem: There is at least one line which is superset of the diagonal. As there is no line with omega elements, there can be no diagonal with omega elements. Regards, WM
From: mueckenh on 20 Nov 2006 09:11 Virgil schrieb: > > If not all digits of the diagonal number exist, then the diagonal > > number is not an example for a number which exists but is not in the > > list. > > But there is nothing that prevents any digit from existing, Existing are such ideas which are accessible. No others are existing, because existence of ideas means existence in the mind of someone. > > > > > > 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 know that at most 10^100 digits of sqrt(2) can be determined, in principle. In praxis there are less digits possible. Therefore the decimal representation of this number can never be treated in a list. Regards, WM
From: mueckenh on 20 Nov 2006 09:21 Virgil schrieb: > 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. Yes. And it is as easy to see that for any character of the diagonal there exists a line which contains this one and the next one. Why do you see only the one side of the medal and dispel that the other side exists too ? Is it because of the joke with the at least one cow which is black at least at one side? Regards, WM
From: Randy Poe on 20 Nov 2006 09:27 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > If not all digits of the diagonal number exist, then the diagonal > > > number is not an example for a number which exists but is not in the > > > list. > > > > But there is nothing that prevents any digit from existing, > > Existing are such ideas which are accessible. No others are existing, > because existence of ideas means existence in the mind of someone. > > > > > > > > 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 know that at most 10^100 digits of sqrt(2) can be determined, in > principle. In principle, if a is the sqrt(2) to 10^100 digits, then 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. What "in principle" prevents me from calculating 2/a, or adding it to 1, or taking 0.5*(a + 2/a)? - Randy
From: Franziska Neugebauer on 20 Nov 2006 09:52
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> A theory of _sets_ is not a theory of _columns_. >> > >> > Experience has shown that practically all notions used in >> > contemporary mathematics can be defined, and their mathematical >> > properties derived, in ZFC. In this sense, the axiomatic set theory >> > serves as a satisfactory foundation for al[l] other branches of >> > mathematics. It can describe every mathematical notion --- with the >> > exception of what a column is? >> >> I do not speculate about what is describable and what not. If you >> want to posit a definition do so! > > "Practically all" notions! If you again refuse to learn from me, look > here: > > Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. In this sense, the axiomatic set theory > serves as a satisfactory foundation for the other branches of > mathematics. [Karel Hrbacek and Thomas Jech: "Introduction to Set > Theory" Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] Again: Meet _your_ obligation and define "column[s]". You have introduces this notion hence it is up to you to define it. >> It is curious that you obviously don't like to give precise >> definitions of certain notions even when you have explicitly been >> asked for. > > I defined the EIT. A column is a vertical row. What is a row in the language of (which?) set theory? > The first "column" of it is the first vertical row. "First" is counted > from the left hand side. Perhaps I should add that "first" means > really first, not zeroth. Would you please posit your claim in coherent sentences in the language of (which?) set theory now? >> [...] >> >> >> > Sometimes it is necessary to quote. In particular if you are >> >> > uninformed but nevertheless refuse to take advice from me. >> >> >> >> I don't need any advice from you. >> > >> > You don't know it. That' s why you would need to learn a lot. Look, >> > you have learned from me meanwhile, even against your furious >> > opposition, that omega = |omega| in modern set theory. >> >> You do no longer persue your plan to prove a contradiction in modern >> set theory? > > Perhaps I will find another proof, but I think those delivered are > sufficient. You have not yet found any. >> > Doesn't this case make you wonder whether there are other things >> > which you do not yet know but which you could learn from me? >> > >> >> My opinion is >> >> >> >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) >> > >> > Seems an empty opinion. What is the symbol A refers to? >> >> A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) [(ISCARD)] > > That is correct. All (initial) segments of natural numbers (which > consist only of natural numbers!) have a finite number of > members. [(WMPHRASING)] I again refer to the definition of initial segment given in http://mathworld.wolfram.com/InitialSegment.html (ISDEF) 1. An initial segment is a special subset of a *set* and not "of [...] numbers" (plural). Your phrasing could lead to the (mis?)interpretation that "segments of natural numbers" means "segments /consisting/ of natural numbers" which is not the point to aim at. Perhaps you could clarify this. 2. Theorem: There is no set which is identical to one of its initial segments. Prove this as a homework and you will learn from yourself. You may discuss the three cases empty set, set with last element and set without last element separately. 3. A better translation of (ISCARD) would read "Every single initial segment of _the_ _set_ of natural numbers has finite cardinality." {0, 1, 2, ..., n} is an initial segment of omega with respect to some element n + 1 e omega. Hence (ISCARD) states that every single initial segment of _the_ _set_ of natural numbers has a finite cardinality. >> What I say is: The cardinality of the sequence (d_nn) is the >> cardinality of omega. > > And the ordinality of (d_nn) is omega too, like the ordinality of the > first column. Whatever ordinality of the column formally is. > Therefore it is clear that all segments of natural numbers (which all > are subsets of the numbers contained in the lines of the EIT) are > finite. (ISCARD) is about initial segements not about plain-vanilla segments which still lack a definition of yours. F. N. -- xyz |