Cantor Confusion
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From: Eckard Blumschein on 5 Dec 2006 09:23 On 12/5/2006 12:34 AM, Virgil wrote: > In article <45745F0A.2000408(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 9:42 PM, Virgil wrote: >> > In article <45706268.1020005(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 12/1/2006 1:37 AM, Dik T. Winter wrote: >> >> > In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein >> > >> >> > Oh, for once, try to talk mathematics. By the axiom of infinity the >> >> > set of all naturals is neither hypothetical nor fictitious. >> >> >> >> This axiom combines flawless Archimedean reasoning with an at least >> >> questionable replacement of the notion number by the notion set. >> > >> > When EB presents a completed axiom system from which he can generate >> > mathematics, or at least arithmetic, he may join the lists, but until >> > then he is merely a spectator at mathematics, and not competent to be a >> > judge. >> >> At least, a spectator is not blind. > > A spectator who chooses not to see what is there to be seen, as WM keeps > doing, is no better off that if he were blind. Why do you mean is to be seen? That there are more rationals than reals? I see it, see it a fallacy behind it, see the consequences of this fallacy carefully hidden behind the huge junk of set theory und hope for more intelligent youngsters.
From: Georg Kreyerhoff on 5 Dec 2006 09:36 Eckard Blumschein schrieb: > Do not confuse Cantor's virtue of belief in god given sets with my power > of abstraction. Your power of abstraction is nonexistant. You're not even able to distinguish between representations of numbers and the abstract concept of numbers. Georg
From: Bob Kolker on 5 Dec 2006 10:10 Eckard Blumschein wrote: > sinde and generally uncountable just fictitious reals on the other side. > > In other words: Genuine numbers are countable, fictitious numbers are > uncountable. The latter do not have an available numerical address. For the latest time: countability is a property of sets, not individual numbers. There is no such thing as a countable integer, countable rational or countable real. Bob Kolker
From: stephen on 5 Dec 2006 10:53 mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > stephen(a)nomail.com schrieb: >> >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > No. X and Y do not grow, they remain "X" and "Y". The set they denote >> >> > does grow. The number of EC states may be n. "n" does not grow. The >> >> > number denoted by "n" does grow. >> >> >> >> What do you mean by the 'the number denoted by "n" does grow'? >> >> Currently the number of EC states is 25. In a month it will be 27. >> >> Does that mean 25 is going to grow into 27? Will 25 no longer exist? >> >> Or will 25 now mean 27? What do you mean by 'the number denoted by "25" does >> >> grow'? >> >> > It is only a matter of definition and in principle no reason for >> > quarrel. But it is amusing to see ho set theorists insist on the >> > complete and actual existence of the sets of numbers. Of course 25 will >> > not switch to 27 but the number of states will switch from 25 to 27. >> > That's all. Only by this notion we can talk of growing sets and >> > introduce the notion of potential infinity. >> >> So the number denoted by "n" does not grow? You seem to be switching your position. >> >> >> >> >> >> The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. >> >> The idea that a set ever changes is equally ridiculous. >> >> > No. Compare Fraenkel et all. They talk about to look at the universe of >> > all sets not as a fixed entity but as an entity capable of "growing". >> > What they understand and how this growing can take place has lead to >> > many misunderstandings by underinformed mathematicians. But however one >> > may interpret their sentence. The universe of all sets can change, to >> > put it cautiously. That is not at all ridiculous. >> >> > Regards, WM >> >> Nobody but you has talked about "growing" sets. Sets, like numbers, do not >> grow. You, like many other people who do not understand set theory, > Do you really think that there are people who do not understand set > theory (if they try)? > Do you need this conviction for your self-respect? > Regards, WM If you think sets grow, then you do not understand set theory. You simply do not understand what sets, as described in set theory, are. It is like the people who think .9999.. approaches 1. They simply do not understand what numbers are. There is nothing wrong with not understanding something. There are lots of things that I do not understand. However I try not to talk about things I do not understand, nor try to correct people whose understanding of a subject is better than mine. Stephen
From: stephen on 5 Dec 2006 10:57
mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > stephen(a)nomail.com schrieb: >> >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > Dik T. Winter schrieb: >> >> >> >> >>Do they define sets as allowed to grow? Not in the quote you supply. >> >> >> There they talk about set valued variables that can grow. >> >> >> >> > No. X and Y do not grow, they remain "X" and "Y". The set they denote >> >> > does grow. The number of EC states may be n. "n" does not grow. The >> >> > number denoted by "n" does grow. >> >> >> >> What do you mean by the 'the number denoted by "n" does grow'? >> >> Currently the number of EC states is 25. In a month it will be 27. >> >> Does that mean 25 is going to grow into 27? Will 25 no longer exist? >> >> Or will 25 now mean 27? What do you mean by 'the number denoted by "25" does >> >> grow'? >> >> > It is only a matter of definition and in principle no reason for >> > quarrel. But it is amusing to see ho set theorists insist on the >> > complete and actual existence of the sets of numbers. Of course 25 will >> > not switch to 27 but the number of states will switch from 25 to 27. >> > That's all. Only by this notion we can talk of growing sets and >> > introduce the notion of potential infinity. >> >> So the number denoted by "n" does not grow? You seem to be switching your position. >> >> >> >> >> >> The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. >> >> The idea that a set ever changes is equally ridiculous. >> >> > No. Compare Fraenkel et all. They talk about to look at the universe of >> > all sets not as a fixed entity but as an entity capable of "growing". >> > What they understand and how this growing can take place has lead to >> > many misunderstandings by underinformed mathematicians. But however one >> > may interpret their sentence. The universe of all sets can change, to >> > put it cautiously. That is not at all ridiculous. >> Nobody but you has talked about "growing" sets. > Of course they did. And if they would not have done so, that would not > at all change the matter. You have failed to provide any citation where somebody talks about sets "growing". As others have pointed out, the citation you have provided does not say what you think it does. >> Sets, like numbers, do not >> grow. You, like many other people who do not understand set theory, >> think of sets as mutable objects, that change as we perform operations >> on them. This is akin to thinking that numbers change when we perform >> addition. If I add 3 to 7, neither 3 or 7 changes. > Please read before answering. I did. It contradicts your earlier statement that the number denoted by "n" grows. No numbers grow, just as no sets grow. >> > It is only a matter of definition and in principle no reason for >> > quarrel. But it is amusing to see ho set theorists insist on the >> > complete and actual existence of the sets of numbers. Of course 25 will >> > not switch to 27 but the number of states will switch from 25 to 27. >> > That's all. Only by this notion we can talk of growing sets and >> > introduce the notion of potential infinity. Yes, 25 will not switch to 27, and {1,2,3} will not switch to {1,2,3,4,5}. But there are no growing sets. The membership of each set is fixed and unchanging. Stephen |