Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: William Hughes on 5 Dec 2006 08:44 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > > We extend this to potentially infinite sets: > > > > > > > > > > > > A function from the set potentially infinite set A to the > > > > > > potentially infinite set B is a potentially infinite set of > > > > > > ordered pairs (a,b) such that a is an element of A and b is > > > > > > an element of B. > > > > > > > > > > > > We can now define bijections on potentially infinite sets > > > > > > and extend the bijection equivalence relation to include > > > > > > potentially infinite sets. Thus we can define > > > > > > equivalence classes under bijection of potentially infinite sets. > > > > > > Thus we can define "cardinal numbers" of potentially > > > > > > infinite sets. > > > > > > > > > > > There is only one "cardinal number". In order to apply any of Cantor's > > > > > proofs of higher cardinal numbers, a set of aleph_0 must be complete. > > > > > But it cannot be complete in potential infinity. > > > > > > > > You now agree that a potentially infinite set can have > > > > a cardinal number and that this cardinal is not > > > > a natural number. > > > > > > > I wrote: a "cardinal number". oo is not a cardinal number in the sense > > > of set theory. > > > > > > > We have: there exists a bijection between sets or potentially infinite > > > > sets > > > > A and B iff the cardinal number of A is the same as > > > > the cardinal number of B. > > > > > > > > Now apply this. > > > > > > > > The natural numbers form a potentially > > > > infinite set. The diagonal contains the potentially infinite set > > > > of natural numbers. > > > > > > There is nothing to contain! You are too much caught in the terms of > > > set theory. You cannot have the complete set because then it would be > > > complete, i.e., actually existing, i.e., actually infinite. > > > > > > If you want to avoid the word contain, reword > > "The diagonal contains the potentially infinite set > > of natural numbers." as "if x is an element of the potentially > > infinite set of natural numbers, then x is an element of > > the potentially infinite set of elements of the diagonal". > > And as well: If x is an element of the potentially infinite set of > natural numbers, then x is an element of a line (and all natural > numbers y < x are also elements of that very line). > > And there is no natural number x, which is no element of a line. Let us write this out, taking care to note that line n contains all the elements of any line less than n. We have For every natural number n there exists a line L(n), such that every natural number m <= n is an element of L(n) You cannot simply exhange the quantifiers to get There exists a line L, such that for every natural number n, every natural number m<=n, is contained in L. You need to give some other argument to show that L exists. This you have not done. It is in fact easy to show that L cannot exist. If X is such that for every natural number n, n is an element of X, then X is a potentially infinite set. No line L is a potentially infinite set. Therefore, there does not exist a line L such that for every natural number n, every natural number m <= n is an element of L. The fact that there each natural number is a member *some* line, does not mean there is one line which contains all natural numbers. The point remains. A set with a largest element can have elements all of which are natural numbers. A potentially infinite set without a largest element can have elements all of which are natural numbers. There cannot be a bijection between a set with a largest element and a potentially infinite set without a largest element. Therefore there cannot be a bijection between the diagonal and a line. Do you intend to keep claiming that a bijection can exist between the diagonal and a line?. - William Hughes
From: Eckard Blumschein on 5 Dec 2006 08:49 On 12/5/2006 1:14 AM, Virgil wrote: > In article <457467D5.7020201(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 8:55 PM, Virgil wrote: >> > In article <45700723.3060406(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 11/30/2006 1:39 PM, Bob Kolker wrote: >> >> >> > Division by zero in a field yeilds a contradiction. >> >> >> >> Just this contradiction resides already in the notion of (actual) >> >> infinity. >> > >> >> > Division by zero in standard sets of numbers is not defined because >> > there is never a unique x in such sets of numbers for which a = 0*x. >> > Either no x works or more than one works. >> > >> > Infinity has nothing to do with it. >> > >> > A finite example: >> > >> > The residues of the integers modulus a prime is always a finite field >> > under the usual addition and multiplication, so there is no >> > "infinity" involved, but division by zero in those fields is still >> > barred for the reason above, a = 0*x can never have a unique solution. >> >> I do not feel limited in thinking to the indefinitely large. I likewise >> consider the indefinitely small (infinitesimal). > > In finite rings, both are irrelevant, but the issue of division by zero > is the same even in such rings. Those who try to drag in the infinite or > infinitesimal in discussing the division by zero issue, just do not > understand the issue. Hopefully you can substantiate this pure suspicion. Being an engineer, I vaguely recall that a Zahlring is something like a loop. Let me fantasize: {i, i^2, i^3} Is this a ring? So far I do indeed not understand why the issue of rings matters in case of division by zero. >> >> Isn't it better to understand why it is incorrect than simply to learn >> >> it is forbidden? >> >> >> >> Eckard Blumschein >> > >> > It is better to understand the real reason (see above), but Eckard >> > doesn't seem to understand the real reason. It has nothing to do with >> > "infinity". >> >> Not directly with the indefinitely large, yes. > > Where does the "infinitely large" or "infinitesmially small" enter into > finite rings, such as the fields of integers modulo a prime? I do not grasp your point. Mathematical closed loops (meshs) are of course pathways of infinite length. Correspondingly stars (nodes) add all branches to the indefinitely small (zero). > > The division by zero question has the same answer, and for the same > reasons, in these rings as in infinite rings. Why not?
From: Dik T. Winter on 5 Dec 2006 08:42 In article <457542CF.2070008(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > 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: .... > > > 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. In languages words change meaning over time. -- 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 5 Dec 2006 08:45 In article <4575597D.3050903(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 12/5/2006 2:50 AM, Dik T. Winter wrote: .... > > In the one-point compactification of the real line or the complex plane > > a single point at infinity is added. But also in that case it is either > > a real nor a complex number. And again, the result is not a field, and > > also not odered. > > Obviously, you meant not either but neither? Indeed. > I wonder why the mathematicians believe to require one-point > compactification. I consider the rationals as genuine numbers, being as > close as you like to the fictions infinity and real numbers. The exact > numerical representation of pi requires the fiction of actual infinity. I wonder why you are talking about things you know nothing about? Who requires one-point compactification with what goal? -- 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 09:19
On 12/5/2006 12:39 AM, Virgil wrote: >> If irrational numbers are thought >> to complete the rationals which sounds quite logical, then the >> constituted entity of the reals has to be as fictitious as the >> irrationals. > > And the rationals and the naturals and all other mathematical > constructions. No. No. No. I do not have a more apt word than fictitious in the sense it was used by Leibniz in order to stress the conceptual difference between addressable discrete numbers and merely attributed without such address positions. If you deny this conceptual difference, then you are denying the difference between generally countable rationals on one 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. > They are all equally fictitious, creations of the mind having no > existence outside of the mind. This is an attempt to hide that Dedekind and Cantor built an Utopia. |