Cantor Confusion
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From: David Marcus on 6 Dec 2006 02:48 Eckard Blumschein wrote: > On 11/22/2006 12:19 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Every set of natural numbers has a superset of natural numbers which is > >> > finite. Every! > >> > >> I am only aware of the unique natural numbers. If one imagines them > >> altogether like a set, then this bag may also be the only lonly one. > > > > WM means that every finite set of natural numbers is contained in a > > larger finite set of natural numbers. > > The sequence of natural numbers has no reason to end, exept if one does > not have enough power of imagination or abstraction, respectively, to > exclude any physical restriction from the ideal model of natural numbers. > > > Since he believes all sets of > > natural numbers are finite, he then concludes that every set of natural > > numbers is contained in a finite set of natural numbers. This seems to > > be due to an allergy to the word "set". > > Thank you for clarification. > > The word set does indeed serve as a deliberately obscuring crutch. Only for cranks, who seem to latch onto the word "set" as signifying more than it does. > It suggests an aprioric point of view, contrasting to Archimedes. > WM denies this selfcontradictory abstraction. > Following Leibniz, I consider uncountable numbers a useful fiction with > a fundamentum in re. You really should learn to stop talking in such vague terms. Why don't you see if you can go a week without quoting a dead mathematician or stating what you "consider"? -- David Marcus
From: David Marcus on 6 Dec 2006 03:00 Eckard Blumschein wrote: > On 11/25/2006 9:50 PM, Ross A. Finlayson wrote: > > > McGill's proof checker found the rationals uncountable. > > Can you point us to an available source, please? Are you really trying to ask Ross a question? > I guess, the mathematical object under test was not the rationals but > the set of rationals understood like a whole entity. > > If so, did the checker also test the naturals? > I see the naturals countable while the entity of all naturals an > uncountable fiction. > > My notion of the reals is different from Dedekind's. > Already each single out of these reals is uncountable. Your notion of everything in math is different from everyone's. How come you didn't make up your own version of English, too? -- David Marcus
From: David Marcus on 6 Dec 2006 03:12 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > It is asserted in that paper that the so called first Cantor proof of > > the uncountability of the reals applies equally well to the set of > > rationals. > > > > Cantor's proof relies on the fact that if one has a strictly increasing > > sequence of reals with each term less than all terms of a strictly > > decreasing sequence of reals, there is at least one real strictly > > between the two sequences, not being a member of either but larger than > > every member of the increasing sequence and smaller that every member of > > the decreasing sequence. > > Ok. This is what I wrote when describing Cantor's proof. > > > > This is not the case for rationals. For example, there are strictly > > increasing sequences of rationals whose squares converge to 2 and > > strictly decreasing sequences of rationals whose squares converge to 2. > > But there is no rational number between the sequences. > > > > For example a_1 =3D a, a_{n+1} =3D 4*a_n/(2 + a_n ^ 2) is an increasing > > sequence of rationals and b_1 =3D 2, b_{n+1} =3D (2 + b_n ^ 2) / (2 * b_= > n) > > is a decreasing sequence of rationals with only sqrt(2) between all the > > a_n and all the b_n. > > Nevertheless the rational numbers and the irrational algebraic numbers > are countable. So, what does the proof, as you described it, show? The > uncountability of a countable set? Where is the error of mine? > > In addition, two sequences of transcendental numbers can converge to a > rational number, such that we have the same situation as described > above. > > Therefore both sets, Q =EF=80=A0and T (transcendental numbers), have the sa= > me > status with respect to *this* uncountability proof. And we are not > able, *based on this very proof*, to distinguish between them. > On the other hand, the proof can show the uncountability of a countable > set. If, for instance, the alternating harmonic sequence > (-1)^n/ n --> 0 > is taken as sequence (1), yielding the intervals (-1 , 1/2), (-1/3 , > 1/4), ... we find that > its limit 0 does not belong to the sequence, although the set of > numbers involved is obviously > denumerable. > The alternating harmonic sequence does not, of course, contain all real > numbers, but this simple example demonstrates that Cantor's first proof > is not conclusive. *Based upon this proof alone*, the uncountability of > this and every other alternating convergent sequence must be claimed. > Only from some other information we know their countability (as well as > that of Q), but how can we exclude that some other information, not yet > available, in the future will show the countability of Q or T? > > > Not so, as the example above proves. There is an increasing sequence of > > rationals converging to sqrt(2) and a decreasing sequence of rationals > > converging to sqrt(2) such that there is no rational at all caught > > between the two sequences. > > Nevertheless the algebraic numbers are countable. > > > > No, it is you who misunderstand Cantor's proof. > > Where is an error of mine? I only see that you understood that the > algebraic numbers are uncountable. > > > > > I claim that Cantor's proof is > > > "symmetric". It can be applied to the algebraic numbers, showing that > > > the limit is not algebraic. > > > > It can be but need not be. > > That is the same with the rational sequences. Some converge to 0. > > > > It can be applied to any proper subset of the reals showing that there > > is a sequence of values in that subset which does not converge to a > > value in that subset. > > Cantor's proof can be used to show the uncountability of the rational > numbers. In Q there are sequences which converge to rational numbers. > > > > >This > > > has nothing to do with rational numbers and sqrt(2) because that would > > > not prove any uncountability at all (sqrt(2) is algebraic and as such > > > belongs to a countable set). > > > > The point is that for every sequence of reals, there are reals not in > > that sequence. > > And for every sequence of algebraic numbers converging to an algebraic > number, there are algebraic numbers not in that sequence.=20 > > Regards, WM Hey, this post is pretty good! WM is close enough to being coherent here to see that he really does have problems with quantifiers. He really seems to be confused about whether the proof has to show something for all lists or for just one list. -- David Marcus
From: David Marcus on 6 Dec 2006 03:19 Virgil wrote: > In article <4575B727.6070006(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > On 12/5/2006 3:36 PM, Georg Kreyerhoff wrote: > > > 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 > > > > Really? > > Really! Really! -- David Marcus
From: Eckard Blumschein on 6 Dec 2006 03:42
On 12/5/2006 8:52 PM, Virgil wrote: > In article <1165322064.705072.182240(a)80g2000cwy.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > >> The number of your contributions has increased by 1 with your post I >> just answer. The same holds for the set of your contributions. > > Such time dependent "sets" are not the same as sets under the rubric of > set theories, as they do not, for example, obey the axiom of > extensionality. Fraenkel 1923, p.190: "Dieses Axiom besagt, dass eine Menge m als vollst�ndig festgelegt gilt, sobald bestimmt ist, welche Elemente in ihr enthalten sind." |