An uncountable countable set
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From: MoeBlee on 26 Aug 2006 18:36 Tony Orlow wrote: > > That's a question for philosophy. I have my own views on what > > mathematics is, but I don't hold that my own concept of mathematics is > > definitive. > > > > Do you have an opinion you'd care to express? I have some views, considerations, biases, and predilections that add up to something of a tentative opinion. I should work on composing some kind of post about it. It's a very broad question, and I don't want to loose sight of the fact that there are many aspects, ranging from pure mathematics to applied mathematics, from mathematics in abstraction to human mathematical activities, etc. MoeBlee
From: MoeBlee on 26 Aug 2006 18:43 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Yes, and the universe is consistent by definition, so math should be > >> consistently overall as well. > > > > Unless the universe is a set of sentences, the notion of consistency > > does not even apply. > > The universe is governed by the properties of the elements within it, > which properties are statements true about those elements. You are taking properties to BE statements. That's fine if you have some coherent philosophy to support it. MoeBlee
From: imaginatorium on 26 Aug 2006 23:31 MoeBlee wrote: > Tony Orlow wrote: > > Dik T. Winter wrote: > > > Tony Orlow wrote: > > > > I don't disagree that the first is infinite while the second is > > > > unbounded bt finite, and therefore smaller. > > > > > > Pray, for once, provide a definition. A set is either finite or infinite. > > > And if a set is finite, by the definitions there is a largest element. > > > So, how do you define "unbounded but finite"? > > > > Not with the Dedekind definition. A truly infinite set must have > > elements infinitely many position beyond other elements, the way I see it. > > The man is PLEADING with you for a definition, and you give him > circularity: > > "A truly INFINITE set must have elements INFINITELY many position > beyond other elements" [all caps added] > > And this circularity has been pointed out to you time and time again. > So OF COURSE people get fed up with your twaddle. You can say that again... Brian Chandler http://imaginatorium.org
From: mueckenh on 27 Aug 2006 10:28 Dik T. Winter schrieb: > In article <44eef4aa(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > ... > > The set of all naturals numbers consists of only natural numbers. There > > is NO natural number where the count becomes infinite. So there is no > > point in the set, even if you COULD get to the "last" one, where any > > infinite set has been achieved. > > And there is no point in the set where you have the complete set. Yes. > Indeed. So what? And there is no such point outside either. That is why the complete number 0.111... does not exist. It would be the complete set, it would be the end of the set. It would mean a number larger than any natural number. Regards, WM
From: mueckenh on 27 Aug 2006 10:32
Dik T. Winter schrieb: > > > The first part of his first proof shows that a complete ordered field > > > has cardinality larger than the natural numbers. In his proof he did > > > not rely an any properties of the reals other than that they form a > > > complete ordered field (he uses reals to exemplify). > > > > "Wenn eine nach irgendeinem Gesetze gegebene unendliche Reihe > > voneinander verschiedener reeller Zahlgr=F6=DFen .... vorliegt, so l=E4=DFt > > sich in jedem vorgegeben Intervalle ...eine Zahl ... (und folglich > > unendlich viele solcher Zahlen) bestimmen, welche in der Reihe (4) > > nicht vorkommt; dies soll nun bewiesen werden. > > Can you read German? Reelle Zahl, Zahlgr=F6=DFe, Zahl, Zahl, Zahl. 10 > > times "Zahl" appears in this paper. > > Note what I wrote: "in his proof he did not rely on any properties of > the reals other than that they form a complete ordered field". What > other properties of the reals did he use? He had no field and he used no filed. What is a field without the axioms of the field? Nothing. Cantor disliked axioms if he did not hate them. His only concern were numbers, numbers, numbers and their *reality*. No fields and no axioms. > > > Cantor uses neither "field" nor "ordered" nor "complete". > > Yes. Pray read again what I wrote. > > > > Cantor's one and only purpose was a proof of the theorem: > > > There are sets with cardinality larger than that of the natural numbers. > > > > He wrote that it was a simpler proof for his first theorem, i.e. the > > theorem that the reals are uncountable. > > And he did *not* write that. Aus dem in § 2 Bewiesenen folgt nämlich ohne weiteres, daß beispielsweise die Gesamtheit aller reellen Zahlen eines beliebigen Intervalles sich nicht in der Reihenform...darstellen läßt. Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist. Obviously his first theorem, "jener Satz", was *not* independent of the irrational numbers. Or why did he stress that his second proof was independent of them? Regards, WM |