Cantor Confusion
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From: mueckenh on 25 Oct 2006 16:12 Virgil schrieb: > In article <1161683454.085728.22020(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Therefore we write the limit. The limit of all n is omega. > > By what definition of "limit"? > > > But the *facts* are based on the number of transactions. The trick with > > the vase is simply to force the infinite number of transaction into a > > finite time interval. > > The only relevant question is "According to the rules set up in the > vase problem, is each ball which is inserted into the vase before noon > also removed from the vase before noon?" > > An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the problem. > > How does Mueckenheim answer? You see my name written above. My first name is Wolfgang, my short sign is WM. Unless you switch to one of these alternatives, this is my last answer to you - be sure. All the balls have been removed before noon. But more balls are in the vase. Conclusion: It is nonsense to talk about the complete set N and about any infinite set, no matter, whether it is done in English or in any formal language. I understand that many of those mathematicians who spent years to study set theory, are upset because of the wasted time, and tend to bad behaviour. > > > Come on, real numbers do exist. > >Where? >The same place that naturals, ordinals and rationals exist, in the mind. >And nowhere else. Correct. And therefore no such thing can exist unless it exists in the mind. But we know that there is no well order of the reals. in any mind, because it is proven non-definable. Nevertheless some cranks insist, it would exist somewhere. Do you know where? Regards, WM
From: MoeBlee on 25 Oct 2006 16:14 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 Who said anytying about talking about "every thing"? > the fact that it does not exist No, we don't need to hold that there exists a set of all sets to prove, in a given theory, that there does not exist a set of all sets. > and that we > cannot have any infinity unless we have the axiom of infinity. To prove there exists an infinite set requires some axioms. If the axiom of infinity is not one of them, then, to prove the existence of an infinite, there has to be some other axiom or axioms aside from the axioms of Z without the axiom of infinity. MoeBlee
From: MoeBlee on 25 Oct 2006 16:32 Lester Zick wrote: > "Definition" in mathematics would appear to mean pretty much whatever > people who claim to do mathematics say it means. No, in mathematical logic we prove that there are certain definitional forms that meet the criteria of elminability and non-creativity. > Ex: Cardinality is > card(x)= least ordinal(x) equinumerous(x). Where did you read that definition? It's nonsense. The definition I gave, which is not nonsense, is: card(x) = the least ordinal equinumerous with x. MoeBlee
From: Sebastian Holzmann on 25 Oct 2006 16:59 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > I am afraid, it will be useless for you to consult any books. > Nevertheless, here is my attempt to teach you some real logic: > Vor allem ist die Bildung der wichtigsten Klasse paradoxer Mengen, > n?mlich 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 beschr?nkterer Mengen durch Aussonderung bzw. > Auswahl, oder die Bildung von Mengen, die in eng umschriebenem Ma? > sozusagen umfassender sind, durch Paarung, Vereinigung, Potenzierung > usw. Mathematics has advanced beyond the stages of the 19th century.
From: Sebastian Holzmann on 25 Oct 2006 17:01
mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > 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, n?mlich 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 beschr?nkterer > Mengen durch Aussonderung bzw. Auswahl, oder die Bildung von Mengen, > die in eng umschriebenem Ma? sozusagen umfassender sind, durch > Paarung, Vereinigung, Potenzierung usw. Please do not quote scholars from some remote time on me. Rest assured that I try not to sound as foolish as you do. |