Cantor Confusion
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From: mueckenh on 25 Oct 2006 15:55 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > > Example: I will use the term "gleeb" to refer to an integer > > which is divisible by pi. > > > > This statement is wrong. And it is nonsense. And don't come up with the > > empty set. > > This statement is not wrong. The statement "There exists a gleeb" would > be wrong. Aren't you allowed to speak about non-existing entities? Some logic desired? You can say "this is no gleeb" but you are wrong to say "let that integer be a gleeb", because that integer does not exist. So it cannot "be". So wenig sich in der Arithmetik der endlichen Anzahlen andere Grundgesetze aufstellen lassen, als die seit Alters her an den Zahlen 1,2,3,... erkannten, ebensowenig ist eine Abweichung von den arithmetischen Grundwahrheiten im Gebiete des Transfiniten möglich. "Hypothesen" welche gegen diese Grundwahrheiten verstoßen, sind ebenso falsch und widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. (Cantor) Regards, WM
From: mueckenh on 25 Oct 2006 15:58 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Sebastian Holzmann schrieb: > >> Oh, I think I begin to see your problem here. But before we can speak of > >> ZFC as a theory, we must first have some sort of "background set theory" > >> available. And if we do not allow that background theory to "have" > >> infinite sets (in some naive way), we cannot even formulate Z, because > >> it consist of infinitely many sentences... > > > > So in principle we need the set of all sets in order to talk about > > every thing including the fact that it does not exist and that we > > cannot have any infinity unless we have the axiom of infinity. Yes, > > some very naive (formalized?) opinion. > > We need something to talk about before we can talk about it. If you But we need certainly nothing infinite, because all our talk is finite. > want, you can call the "background sets" not "sets" but "gleeb" to avoid > misunderstandings. Of course these entities are no sets in the sense of set theory. > And, please, do educate yourself on what you are > talking about. Do you in fact consider this the best way of hiding your "no-ledge". Here I repeat some some education for you (by Fraenkel, that is one of those scholars who made ZF): Vor allem ist die Bildung der wichtigsten Klasse paradoxer Mengen, nämlich der allzu umfassenden Mengen (Antinomien von Burali-Forti, Russell usw.) durch unsere Axiome ausgeschlossen. Denn diese gestatten, eine oder mehrere gegebene Mengen als Ausgangspunkt nehmend, nur entweder die Bildung beschränkterer Mengen durch Aussonderung bzw. Auswahl, oder die Bildung von Mengen, die in eng umschriebenem Maß sozusagen umfassender sind, durch Paarung, Vereinigung, Potenzierung usw. Regards, WM
From: mueckenh on 25 Oct 2006 15:59 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > One can prove the nonexistence, because there are no sets possible > > which cannot be constructed from the empty set. Otherwise you could > > also assert that the set of all sets was in your theory. Just in oder > > to avoid that, the axioms were made. > [...] > > So we cannot be sure in ZFC that the set of all sets does not exist? Or > > what is the difference? > > The existence set of all sets contradicts with the other axioms. One > cannot have a consistent theory that combines ZF(C) with "\exists x > \forall y y\in x". > > One can combine the axiom of infinity with the other axioms of ZF(C) > without getting a contradiction (provided there wasn't one inside ZF - > AoI). That is the difference! One cannot get an infinite set without the axiom of infinity. That is fact! Here you can read again, why the axioms have been made as they are. If you read it for the third time, perhaps you will understand? Denn diese [Axiome] gestatten, eine oder mehrere gegebene Mengen als Ausgangspunkt nehmend, nur entweder die Bildung beschränkterer Mengen durch Aussonderung bzw. Auswahl, oder die Bildung von Mengen, die in eng umschriebenem Maß sozusagen umfassender sind, durch Paarung, Vereinigung, Potenzierung usw. Regards, WM
From: Virgil on 25 Oct 2006 16:02 In article <1161806006.602919.26360(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Oh no, not that! It is enough to know that the subject of what you call > "logic" is nonsense. And we know already that what "Mueckenh" calls logic is nonsense, so we are home free.
From: mueckenh on 25 Oct 2006 16:07
Han de Bruijn schrieb: > David Marcus wrote: > > > Han de Bruijn wrote: > > > >>And no, binary trees will not be found in Halmos' "Naive Set Theory". > >>Because it's too naive, I suppose .. > > > > I don't see how you would know it is not there since you said that while > > you own the book, you've never read it. Funny how books aren't much use > > if you don't actually read them. > > > > Anyway, I didn't say binary trees were in the book. I said that if > > Muckenheim wants to talk using the mathematical concepts of sets, > > functions, and relations, he should use standard terminology, where > > "standard" means the same as in some mathematics book. There are many books on binary trees and many others on geometric series. I put these topics together. That is new and not everybody can understand it immediately. You must not be sad on that behalf. Follow just my discussion with those guys who have understood this comparatively simple matter. At the end, you may get it too. > > Or, he could continue to do as Humpty Dumpty did and use his own > > meanings for words without telling people what they are. You must not think that everything you cannot understand is ununderstandable. It is just some new idea. Otherwise it would not be so interesting. You prefer to learn old knowledge from your old books with their old trodden paths? My paths are new! Regards, WM |