Cantor Confusion
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From: Virgil on 18 Oct 2006 15:25 In article <1161159083.695607.296930(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The set is finite, but has not > a largest element. Then the set is not an ordered set. One cannot have a finite ordered set without a largest element. One also cannot have a finite set without being able to tell which objects are members and which are not. At least not in ZF or NBG.
From: Virgil on 18 Oct 2006 15:29 In article <1161159181.827351.41670(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > Wrong. The connection between finite paths and partial sums of edges > > > leads to > > > (1-(1/2)^n+1)/(1 - 1/2) edges per path. And (1-(1/2)^n+1)/(1 - 1/2) <= 4 for all n in N, and lim_{n -->oo} (1-(1/2)^n+1)/(1 - 1/2) = 4.
From: Virgil on 18 Oct 2006 15:32 In article <1161159542.523113.318440(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1161007554.513186.56640(a)i42g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > > The function f(t) = 9t is continuous, because the function 1/9t is > > > > > continuous. > > > > > > > > Yes, but that is not the number of balls in the vase. > > > > > > > For the t-th transaction 9t is the number of balls in the vase. > > > > Let me clarify. WM apparently asserts that if a function g(x) is > > continuous > > at some point, so is 1/g(x). That is (obviously) false. sin(x) is > > continuous at x = 0, while 1/sin(x) is not. > > > I used but a simple definition of the improper limit oo of the function > x which is given by the fact that the function 1/x has the proper > limit 0 (fo x --> oo in both cases). > > > Moreover, when we let t go to infinity, 1/9t is *not* continuous at > > infinity (whatever that may mean). We can only define the limit, > > not the function value. > > We can only define limits in *all* cases concerning the infinite. > Nothing else is possible. Certainly not the continuity that "Mueckenh" claims.
From: Virgil on 18 Oct 2006 15:40 In article <1161159675.931671.301310(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1161008572.469763.93200(a)f16g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > jpalecek(a)web.de schrieb: > > ... > > > > The fact that you cannot compute a list of all computable reals does > > > > not mean that there is no list of all computable numbers. There is > > > > one, > > > > and it is not computable. > > > > > > > The fact that you cannot compute a list of all reals does not mean that > > > there is no list of all reals. There is one, but it is not possible to > > > publish this list. > > > > You are seriously wrong. > > You should have noted that this was an ironic reply. But in order to > avoid machines and undecidabilities: > The set of constructible numbers is countable. Any diagonal number is a > constructed and hence constructible number. No nth-listed number needs to be "completely" constructed, but only constructed far enough to determine its nth digit, so that no number in the listing needs to be constructible, nor does the set of listable numbers need to be a subset of the set of constructible numbers. > > Every list of reals can be shown incomplete in exactly the same way as > every list of contructible reals can be shown incomplete. But as the lists need not be restricted to constructible numbers in order to allow construction of a diagonal different from all of them, that argument does not hold. > > Regards, WM
From: mueckenh on 18 Oct 2006 15:42
MoeBlee schrieb: > MoeBlee wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > You haven't yet noticed it? Each digit of the infinitely many digits of > > > the diagonal number has the same weight or importance for the proof. In > > > mathematics, the weight of the digits of reals is 10^(-n). Infinite > > > sequences of digits with equal weight are undefined and devoid of > > > meaning. > > > > The proof doesn't contradict the fact that the members of the sequence > > are divided by greater and greater powers of ten. That fact is > > mentioned in the previous proof showing the correspondence between the > > sequences and real numbers. We prove that every sequence corresponds to > > a real number where the real number is the limit of the sum of the > > sequence made by taking greater and greater powers of ten in the > > denominators, and that every real number corresponds to such a > > sequence. THEN we proceed to the diagonal argument. > > > P.S. Again, if you disagree with the proof, then please just say what > axiom or rule of inference you reject. In the meantime, again, there is > no rational basis whatsoever for disputing that the argument does > indicate a proof from the axioms per the rules of inference. Neglecting the powers 10^(-n) converts an infinite sequence which can possibly yield a meaningful result into an impossible sequence, which cannot be treated at all. But I believe your intuition will hinder you accept that. Regards, WM |