Cantor Confusion
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From: Dik T. Winter on 4 Dec 2006 21:03 In article <1165228714.543673.34570(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > Therefore we can denote a set by X and we can say that the set X grows. > > > > Wrong, again. Consider X a variable that stands for an unspecified > > integral number. Can you state that the integral number X grows? > > Of course the unspecified number grows by taking the values of > specified numbers. How can a number grow, even when it is unspecified? But of course you can not state that "the integral number X" grows, because X is not a number, it is a variable. > > > That is nothing else than to say that the number of states in the EC > > > grows. Of course the number 6 has not gown to 25. But it is simply a > > > matter of definition, how one interprets "to grow" and "number". > > > > In that case, please provide a definition. > > Everybody knows what the number of ther EC states is. That is *not* what I did ask you. You state that it is simply a matter of definition how one interprets "to grow" and "number", and I asked you to provide definitions. Moreover, the number of the EC states is not fixed, so you can only state what the number of the EC states is at a particular point in time. > > > > That is not "the set of states". You can talk about "the current > > > > set of states" or about "the set of states in 1957" or whatever. > > > > At least mathematically. In mathematics, by definition, a set > > > > can not grow. > > > > > > Wrong. Read my explanation above. > > > > Wrong. Read my explanation above. I am talking mathematics. > > Do you really think so? Yes. > > > > You are, of course, entitled to use another definition, but that will > > > > not clarify the discussion at all (and you are not using standard set > > > > theory). > > > > > > Hrbacek and Jech teach standard set theory including the fact that in > > > ZF everything is a set. > > > > Yes? > > I remember that you opposed. Now you agree because you have learnt? No. I still do not agree. The universe in ZF is *not* a set. > > 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. That makes no sense. How can a number grow? Like in "the square root of 2 is 1.5 for large enough values of 2"? > > > Hrbacek and Jech teach standard set theory including the fact that in > > > ZF everything is a set. > > > > But I see nothing that states that a set can grow. > > Pray reread. If you do not yet understand it reread again. Not found anything that alludes to possibly growing sets. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 4 Dec 2006 21:06 In article <1165242365.580949.112500(a)l12g2000cwl.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > By the axiom it is not assumed that it exists, it is stated that it > > exists. It is similar to the parallel postulate from Euclid that > > does not assume that there is one line going through a point not on > > another line and parallel to that other line. It is stated as fact. > > To have a big mouth is not enough to create a world, not even a notion. > You would see that if you tried to say where the assumed object > existed. In my mind. I can reasonably think about the set of all natural numbers. Honest, I have no problem with it. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 4 Dec 2006 21:10 In article <457434EA.70003(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 12/2/2006 2:56 AM, Dik T. Winter wrote: .... > > By the axiom it is not assumed that it exists, it is stated that it > > exists. It is similar to the parallel postulate from Euclid that > > does not assume that there is one line going through a point not on > > another line and parallel to that other line. It is stated as fact. > > And that gives us Euclidean geometry. In the same way, the axiom of > > infinity gives us ZF set theory where the set of naturals does exist > > as a reality. > > The Latin word factum means something which has been done. Wo created > the factum of set theory? What factum? > I do not refer to the axiom of infinity. This > is just an abused version of the Archimedean original. Nothing more than opinion, not mathematics. > I refer to the > idea that there are more countables than fictitious elements of the > uncountable continuum. Eh? What idea? Pray explain, as such it makes no sense. In ZF there are more elements in the uncountable continuum than there are countables. > Yes, more than 100 years of fruitless confusion > in mathematics go back to Dedekind and Cantor. I think the confusion is all yours. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Eckard Blumschein on 5 Dec 2006 04:58 On 12/5/2006 3:10 AM, Dik T. Winter wrote: > In article <457434EA.70003(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > > On 12/2/2006 2:56 AM, Dik T. Winter wrote: > ... > > > By the axiom it is not assumed that it exists, it is stated that it > > > exists. It is similar to the parallel postulate from Euclid that > > > does not assume that there is one line going through a point not on > > > another line and parallel to that other line. It is stated as fact. > > > And that gives us Euclidean geometry. In the same way, the axiom of > > > infinity gives us ZF set theory where the set of naturals does exist > > > as a reality. > > > > The Latin word factum means something which has been done. Wo created > > the factum of set theory? > > What factum? You wrote: It is stated as fact. > > > I do not refer to the axiom of infinity. This > > is just an abused version of the Archimedean original. > > Nothing more than opinion, not mathematics. The Archimedean original is a golden treasure of mathematics. > > > I refer to the > > idea that there are more countables than fictitious elements of the > > uncountable continuum. > > Eh? What idea? Pray explain, as such it makes no sense. In ZF there > are more elements in the uncountable continuum than there are countables. Dedekind wrote already in 1872 what he thoght since 1858: There must be more numbers than rational numbers because the irrational numbers are additional numbers. Cantor made the same fallacious conclusion. The uncountable continuum does not contain countable elements. Any quantitative comparison must be based on quanta, i.e. numbers. Infinity is not a quantum but a property, olthough Cantor believed infinity to be a firm quantum. > > > Yes, more than 100 years of fruitless confusion > > in mathematics go back to Dedekind and Cantor. > > I think the confusion is all yours. CH, AC, endless worries of students, endless debates everywhere, a ridiculous book foundations of set theory showing that there is no proven basis at all, there is not even a valid definition of a set, ... I am not confused. In Germany there is a thread for some weeks or even months: Alptraum reelle Zahlen
From: Eckard Blumschein on 5 Dec 2006 06:20
On 12/5/2006 3:06 AM, Dik T. Winter wrote: > In article <1165242365.580949.112500(a)l12g2000cwl.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > By the axiom it is not assumed that it exists, it is stated that it > > > exists. It is similar to the parallel postulate from Euclid that > > > does not assume that there is one line going through a point not on > > > another line and parallel to that other line. It is stated as fact. > > > > To have a big mouth is not enough to create a world, not even a notion. > > You would see that if you tried to say where the assumed object > > existed. > > In my mind. I can reasonably think about the set of all natural numbers. > Honest, I have no problem with it. Somewhere I read that it is seemingly simple to imagine the set of all natural numbers. One just needs to be either pretty blueeyed or a pretender. It is as easy as to believe in god. Hilbert in 1925 admitted: Infinity evades our imagination. Nonetheless, it is quite reasonable to operate with the notion of not just indefinitely many natural numbers but with the fiction of all natural numbers. Some too honest and blueeyed mathematicians will perhaps not even understand that there is a categorical difference between the potential and the actual infinite points of view. Those who are rather too flexible will also deny this difference because they feel safe behind the argument that mathematics does not require ultimate correctness, and just convergence matters. |