Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 5 Dec 2006 07:40 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. > 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. > > 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. Regards, WM
From: mueckenh on 5 Dec 2006 07:52 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. > > > 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
From: Eckard Blumschein on 5 Dec 2006 08:07 On 12/5/2006 1:20 AM, Virgil wrote: > In article <45746B98.5040606(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 8:20 PM, Virgil wrote: >> > In article <1164967792.130794.251330(a)j72g2000cwa.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> May be if you apply your personal definition of potentially infinity, >> >> but not if you apply the generally accepted definition. >> > >> > What "generally accepted" meaning is that? Most mathematicians do not >> > accept that a set can be "potentially" infinite without being actually >> > so. >> >> I see it quite differently: Potentially and actually infinite points of >> view mutually exclude each other as do countable and uncountable, >> rational and irrational. > > Except that countable and uncountable coexist within the same set theory > and rational and irrational coexist within the same real umber field. Cantor's DA2 illustrates that there is no such field/list of real numbers. Isn't this "coexistence" on the same low level of abstraction a basic though hard to unveil intentional mistake by Dedekind? Dedekind argued: As naturals can be extended to the integers in order to allow subtraction and include negative numbers, and integers can be extended to rationals in order to allow division and include fractions, so rationals can perhaps be extended to reals in order to allow non-linear operations and include irrationals. Being mislead by the idea of a dotted line of numbers, he overlooked two aspects. First of all, the irrationals cannot be located numerically. Secondly, the irrationals are not an addendum to the reals but the other way round, the reals vanish completely within the sauce of irreals. The irrationals are at best fictitious numbers because they do not have an exact numerical representation available. Kronecker said, they are no numbers at all. Since the properties of the reals have to be the same as these of the irrationals, all reals must necessarily also be uncountable fictions.
From: Eckard Blumschein on 5 Dec 2006 08:13 On 12/5/2006 1:17 AM, Virgil wrote: > In article <45746ACD.1020008(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 8:27 PM, Virgil wrote: >> > In article <45700481.7010300(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> >> Why should mathematics be esoteric? >> > >> > Not all of it is. Various bits of it come at various levels of >> > abstraction, and even children understand the least esoteric bits. >> >> For my feeling, Dedekind and Cantor were lacking power of abstraction. > > From present evidence, they had a great deal more "power of abstraction" > than EB has. Do not confuse Cantor's virtue of belief in god given sets with my power of abstraction. Cantor said: Je le vois, mais je ne le crois pas. Obviously he didn't infinity.
From: Bob Kolker on 5 Dec 2006 08:13
Eckard Blumschein wrote: > an exact numerical representation available. Kronecker said, they are no > numbers at all. Since the properties of the reals have to be the same as > these of the irrationals, all reals must necessarily also be uncountable > fictions. For the latest time. Uncountability is a property of sets, not individual numbers. There is no such thing as an uncountable real number. Nor is there any such thing as a countable integer. Countability /Uncountability are properties of -sets-, not individuals. You have been told this on several occassions and you apparently are too stupid to learn it. Bob Kolker |