Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 2 Nov 2006 19:07 Lester Zick wrote: > On Fri, 03 Nov 2006 02:27:20 +0800, Noehl <n.alinsangan(a)gmail.com> > wrote: > > >Lester Zick wrote: > >> On Thu, 2 Nov 2006 00:28:49 -0500, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >>> Dik T. Winter wrote: > >>>> In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > >>>> > Yours are some of the very > >>>> > few that don't. And I've seen posts of other .nl correspondents which > >>>> > come through in color. > >>>> > >>>> It has nothing to do with the domain where you come from. It has everything > >>>> to do with the manner things are quoted and in the way *your* newsreader is > >>>> able to handle that. I know the reason your newsreader does not show colours. > >> > >> But my newsreader does show colors. > >> > >>>> It is because I prepend the '>' sign with a space when quoting. > >> > >> Ok. Mine reacts the same way. But when there is no preceeding space > >> quoted text is shown in a different color. > >> > >>> And I > >>>> have pretty good reasons to do that. (One of the reasons being that this > >>>> article would be rejected because there is more quotation than new text.) > >> > >> I've never experienced that. > >> > >>> My newsreader lets me post articles that have more quotation than new > >>> text. > >> > >> ~v~~ > > > >Thunderbird quotes nicely > > Yeah, I'm not sure what the problem is if there is a problem. Dik told you the reason. He's putting " >" at the front of each line instead of ">". So, your newsreader won't see it as a quote. He does this because his newsreader tries to enforce etiquette by not letting him post messages that are mostly quotes. You know, how usenet works is less complicated than mathematics. -- David Marcus
From: Virgil on 2 Nov 2006 19:22 In article <1162472271.664965.315880(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > It is unbounded but always finite. > > > > > > > > Unbounded implies not finite, as finite implies bounded. > > > > > > That is the definition of potential infinity. But for this definition > > > we never have a completet set and never can be sure whether a set is > > > Dedeking infinite. > > > > A set which f(x) = x+1 maps to a proper subset is Dedekind infinite. > > If a set is not actually completed, then it cannot be decided whether > it maps to a proper subset. The axiom of infinity in ZF gives us a "completed" set which is Dedekind infinite. > > > > > > Sorry, to say, but you always mix up these two very different things. > > > > A difference which makes no difference is no difference. > > > > The only difference in whether the set of finite naturals is potentially > > or actually infinite is whether one is looking at its members or at the > > set as a whole. > > Interesting that you mean to have better insights than Cantor who > mainly developed this field. We have had another century or so to study the issue, whereas Cantor, as a pioneer, had to stumble his way through it anew. > > > The difference is in what the viewer chooses to see not > > in what is there to be seen. > > That is a true word! In set theory one sees what one wants to see, not > what is really there. In set theory, one sees what the axioms say is there. Or, if one comes on internal contradictions, one goes back and revises the axioms until no such internal contradictions intrude. All of the alleged contradictions that others have claimed of ZF are external, following from assumptions other than the axioms themselves. > > > > To those of us who choose to see both the members and the set, the > > artificial distinction that Mueckenheim chooses to impose does not exist. > > It is not my distinction. Yes it is, as it does not exist in ZF or NBG. > Actual infinity is mainly due to Cantor. With > regard to MatheRealism it does not exist. What are the axioms of "MatheRealism" and where can one find any analysis of it? > > All sorts of people have tried to show that this assumption leads to > > contradictions within those theories, so far without success. > > Of course. As long as the believers believe that a matrix of finite > lines can have an infinite line (which is the projection of the > diagonal) there can't be any success. As long as WM's true believers believe that every set of naturals must have a largest member, as WM's arguments require, they will be lead up the garden path by such fanatics as WM. > Astonishing is only that so many > people decided to join this club. But perhaps they were not aware of > these things when deciding to study set theory. Therefore I'll try to > inform them. Fools are often astonssihed at the foolishness with which the wise reject fools' claims. > > > > WM deliberately blinds himself by assuming that if every string of > characters in an infinite list of strings is finite that there must be > a > UNIFORM limit on their lengths. > > If actual infinity like omega does exist and if it is larger than any > natural number, then omega is a uniform limit of all natural numbers. Which means, in effect, no finite limit at all. If omega is the only limit and LUB, then there must be strings of length greater than any finite natural. For every n in N there must be infinitely many strings of length greater than n. As otherwise there would have to be a finite longest string and a finite LUB. A > limit which is not assumed. It is not a maximum but only a supremum. > This is a very small but nevertheless decisive difference. Except that having omega as supremum means that the are infinitely many strings greater than every finite length. Otherwise one can prove a finite supremum. > > > But a trivial example proves him wrong: > For each n in the infinite completed set of naturals Net f(n) be a > string of length at least n, then there is no maximally long string > in > the image of f, as for each m in N there is n in N with n > > Length(f(m)). > > Correct. There is no maximum. But the supremum omega is *not* taken. > omega is by definition the number of numbers, in a matrix containing > all natural numbers as lines, omega is the number of line. But there is > no line omega. Doesn't have to be to get a diagonal of that length. All it requires is that line n be at least n long. > > > Note that the "diagonal" in my example above MUST be greater than any > finite n, so that either there is a number greater than every finite n > and simultaneously less than omega or WM is wrong again. > > Or the assumption of actual infinity is inconsistent. As this is in ZF and ZF has not been shown to be inconsistent, it is WM who is inconsistent. > > > WM is under the mistaken impression that there must be a longest line in > any list of lines. > > No, there is no maximum. But every line is shorter (has less than omega > elements). The diagonal can be projected on the line lines. But is itself of "length" omega, as that is the "number" of digits it contains. > > > Any "diagonal" for this function will have to be of length greater than > every n in N. > > This shows the inconsistency of ZF and NBG. It shows the inconsistency of WM. > > > Does WM choose to claim that a set of naturals which has no maximum must > still have a finite upper bound? > > You can see the inconsistency of actual infinity best by the following > list > > 1 1 > 2 1,2 > 3 1,2,3 > ... ... > n 1,2,3,...,n > ... ... > omega 1,2,3,... > omega + 1 1,2,3,...,omega > > This is a contradiction. omega cannot count the natural numbers, > because they count themselves: There are n numbers up to number n. > omega is not their maximum (which does not exist) but their supremum > which is not reached. Forcing set having to "count" itself, a form of having a set be a member of itself, is the problem but the ordinals of ZF and NBG do not count themselves, they only count their predecessors. So again the problems all lie in WM's system not in ZF or NBG. > > > Or is it true for such a set S that for every n in S there is an m in S > with m
From: Virgil on 2 Nov 2006 19:26 In article <5501f$4549f151$82a1e228$30874(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel > > Dekker Inc., New York, 1984, 2nd edition, p. 2: "So the only objects > > with which we are concerned from now on are sets." > > Karl Marx, "Das Kapital": because capitalism is only interested in its > "ungeheure Waren-sammlung". And nothing else matters. > > Han de Bruijn Does HdB declare Marx as necessarily correct when Marxism has so obviously been a massive failure?
From: Dik T. Winter on 2 Nov 2006 21:54 In article <j2bkk2hmo8eg2bev1qmt781mf95f2sf3n6(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Thu, 2 Nov 2006 00:28:49 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >Dik T. Winter wrote: > >> It has nothing to do with the domain where you come from. It has > >> everything to do with the manner things are quoted and in the way > >> *your* newsreader is able to handle that. I know the reason your > >> newsreader does not show colours. > > But my newsreader does show colors. Not for my articles apparently. > >> It is because I prepend the '>' sign with a space when quoting. > > Ok. Mine reacts the same way. But when there is no preceeding space > quoted text is shown in a different color. I do not understand the first sentence. What "same way"? But whatever, yes, I know that removing that space from my quotation prefix would give you back your glorious colours. > >> And I > >> have pretty good reasons to do that. (One of the reasons being that this > >> article would be rejected because there is more quotation than new text.) > > I've never experienced that. Oh, well. So be it. You use a different newsreader than I use and you never did use the newsreader I use. > >My newsreader lets me post articles that have more quotation than new > >text. And the same holds for David Marcus. One of the problems of that is that so many users quote a complete article and add "me too" only. One of the main reasons I keep to this newsreader is that it is simple. It gives me access to all functions without using the mouse, just the left hand part of the keyboard. I am probably obliged to re-translate it in the future, but as the source is readily available, that is no problem. When doing so, I *might* remove the check on quoted text vs. new text, but I do not think there is any need. On my black-white display, the preceding space in quotes much more shows that it is a quote. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 2 Nov 2006 22:06
In article <virgil-1E8243.17262302112006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > In article <5501f$4549f151$82a1e228$30874(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > Karl Marx, "Das Kapital": because capitalism is only interested in its > > "ungeheure Waren-sammlung". And nothing else matters. > > Does HdB declare Marx as necessarily correct when Marxism has so > obviously been a massive failure? What you are doing now is quite similar to what other people are doing in this newsgroup. Because some interpretation of the theory has been a massive failure does not, in principle, mean that the theory itself is a failure. Using that reasoning we can state that Mueckenheims interpretation of set theory is a failure, and so set theory is a failure. And I will refrain from politics in the remainder of this thread. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |