Cantor Confusion
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From: Virgil on 26 Oct 2006 14:42 In article <1161851204.836383.105550(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > > Yes, just as I said, the discussion is about the philosophy of > > > > mathematics and set theory (and, I should add, about informal concerns > > > > and motivations), but there is not, WITHIN the set theory discussed > > > > there, a definition of 'actually infinite' and 'potentially infinite'. > > > > > > Does the study of formal languages really make incapable of > > > understanding plain text? What is written above means: "INFINITY" IN > > > SET THEORY IS ALWAYS "ACTUAL INFINITY". > > > > Exactly. There is only one kind of infinity in modern set theory (and > > modern mathematics). We no longer distinguish between "potential > > infinity" and "actual infinity. These distinct notions were important > > historically until the concept of infinity became better understood. > > LOL. By such students as you, i.e., by people who refuse to understand > anything but compete with computers for the most formal formalism? > > Regards, WM Those who assume a system of axioms stating that that system is merely assumption understand what they are doing. Those like "Mueckenheim" who declare their own,infallibility are doomed to be in error.
From: Virgil on 26 Oct 2006 14:49 In article <1161851714.847818.24460(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > Binary Tree > > > > Unfortunately, it was described in a way that I can't understand it. A > > > > wild guess on my part is that you mean to set up a correspondence > > > > between edges and sets of paths. > > > > > > I am sorry, but if you need a wild guess to understand this text, then > > > we should better finish discussion. Observe just how the discussion > > > runs with all those who understood it, like Han, William, jpale. > > > > Han doesn't understand it (although he probably thinks he does). William > > and jpale simply pick the mathematically meaningful statement that is > > closest to what you write and go from there. > > The model is that simple that any student in the first semester could > understand it. Every paths which branches into two paths necessarily > needs two additional edges for this sake. If one has an infinite path and one makes it branch into two infinite paths, one needs infinitely many new edges. But in trying to inject the set of paths to the set of edges, or vice versa, that is irrelevant. > It is only your formalistic > attitude that blocks your understanding. But you must not think that > anybody is blocked like you. It is the fact that it is trivial to inject the set of edges into the set of paths and provably impossible to do the reverse that "blocks" all reasonable people from seeing otherwise. > > > I'd rather wait until you say something coherent before I > > comment on it. > > You mean, until you are able to *recognize* something coherent? In "Mueckenh"'s incoherent rambilngs, excatly so. > > Regards, WM
From: Virgil on 26 Oct 2006 14:54 In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Within the *real* numbers the limit does exist. And a decimal number is > > nothing more nor less than a representative of an equivalence classes. > > So we are agian at this point: The real numbers do exist. For the real > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > is no application of Cantor's argument. Non sequitur. > > > > Where in the construction above did I use the limit omega? > > In using any infinite sequence already WM may need to use omega in analysing infinite sequences , but to everyone else it is merely a matter of delta-epsilonics. > > > > No. By the theory, each decimal number is a representative of an > > equivalence class of sequences of rational numbers. By the construction > > we get another decimal number that is also a representative of an > > equivalence class of sequences of rational numbers. No omega is needed. > > Why then do you think omega is needed at all in mathematics? For infinite sequences, it is not used at all.
From: georgie on 26 Oct 2006 14:54 David Marcus wrote: > georgie wrote: > > David Marcus wrote: > > > georgie wrote: > > > > David Marcus wrote: > > > > > georgie wrote: > > > > > The OP said that a definition was invalid because it was self- > > > > > referential. However, there is no rule against self-reference in modern > > > > > mathematics, so the OP's objection is not valid. > > > > > > > > So the set of sets containing themselves is ok by you. > > > > > > I don't see where I said that. I said there is no rule against self- > > > reference. The rules are specified by the axioms of ZFC. > > > > So it's valid or it isn't and it's only valid when you like it. Is > > that your position. From your reactions it appears that way. If you don't > > explain why you think self-reference is ok in some cases and not others, you > > lose credibility. > > Please read what I wrote: The rules are specified by the axioms of ZFC. > In other words, the axioms tell you when it is allowed. If you take some > math classes or read some books, you could learn the rules. I take it that means your position is that you don't know and want me to tell you because you can't read books.
From: David Marcus on 26 Oct 2006 14:57
Han de Bruijn wrote: > Newtonian mechanics at least raises the _suggestion_ of being completely > deterministic. But I guess it has been Laplace who formulated that idea > explicitly. Anyway, there is _no_ modern physicist who still believes in > in determinism these days. It is amazing that you would post stuff that a quick Web search would show is wrong. Or, you could try reading Physics Today. -- David Marcus |