An uncountable countable set
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From: Virgil on 28 Jun 2006 13:50 In article <1151506772.409678.115720(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > And each time I implement the diagonal number K(f) into the list f > > > > > (insert it before the first ordinary list number, for instance), you > > > > > will construct another diagonal number K(g) and say, look here, this > > > > > diagonal is not contained in your g. > > > > > > > > As that K(g) was not in M(f) , that is irrelevant. > > > > > > That the diagonal of Cantor's list is not in the list is irrelevant. > > > > That is the one thing about Cantor's anti-diagonal number that IS > > relevant. > > That has turned out wrong. If so, the error is entrely "mueckenh"'s. When we attempt to prove that a set A is countable by constructing a surjective function from N to S, "muecken"'s response is that only his functions are allowed. But that contradicts the definition that ANY such surjection establishes countability. There is nothing in that definition which allows "mueckenh" to veto any proposed function. Similarly, in proving that there are no surjections from any set S to its power set P(S), "Muecken" takes unreasoning exception to the statement that , for any f: S --> P(S), the set K(f) = {x in S : x not in f(x)} is a well defined set not in the image of f.
From: mueckenh on 29 Jun 2006 05:19 Virgil schrieb: > > > The non-empty set of positive rationals has no first element, > > > since for every positive rational, 1/2 of it is strictly between > > > it and zero. > > > > This shows that set theory is self contradictory, because the following > > mapping does not destroy well-order but establishes order by magnitude. > > > > > > For n = 1 to oo: > > (q_2n-1, q_2n) --> (q_2n-1, q_2n) if q_2n-1 < q_2n, else: (q_2n-1, > > q_2n) --> (q_2n, q_2n-1) > > For n = 1 to oo: > > (q_2n, q_2n+1) --> (q_2n, q_2n+1) if q_2n < q_2n+1, else: (q_2n, > > q_2n+1) --> (q_2n+1, q_2n) > > And again: > > For n = 1 to oo: > > (q_2n-1, q_2n) --> (q_2n-1, q_2n) if q_2n-1 < q_2n, else: (q_2n-1, > > q_2n) --> (q_2n, q_2n-1) > > For n = 1 to oo: > > (q_2n, q_2n+1) --> (q_2n, q_2n+1) if q_2n < q_2n+1, else: (q_2n, > > q_2n+1) --> (q_2n+1, q_2n) > > ... > > repeat aleph_0 times. > > Where is your proof that it takes the rationals in some given > sequential order and ever produces an ordering by magnitude? If actual infinity exists, then my mapping comes to an actual end. If not, then Cantor's diagonal can never be shown completed and different from all numbers of the list. > > As there is no such thing as a smallest (most negative) rational , how Let's use only the positive rationals, but the problem is the same, because there is no smallest one > does all of your transposing produce what does not exist? How does sqrt(2) = m/n show that sqrt(2) is rational? It does not show that but it shows that sqrt(2) does not exist as a rational. My proof shows that infinity can never be completed. > > As far as I can see what you have is sort of like an infinite bubble > sort, with large values being bubbled upwards past smaller ones, one at > a time, until they bump into an even larger value. > > But at no stage of this operation does there ever get to be infinitely > many values between two other values. There are not infinitely many values between two values, because the set is and remains well-ordered, i.e. each element is indexed by a natural number. And the difference between two naturals is never infinite. > > For example, in the original sequence there are at most finitely many > values between 0 and 1. You are claiming that there will be some single > transposition which makes the up until then finite number of values > between 0 and 1 in one step into an infinite number of values between 0 > and 1. See above. Between two naturals in natural order, there are never infinitely many naturals. Regards, WM
From: mueckenh on 29 Jun 2006 05:28 Virgil schrieb: > > I recently proved that there is no Almighty (stone, force). > > That alleged "proof" is no more valid than your alleged mathematical > proofs. That proof (I am not the original author) shows that is impossible to simultaneously exert two Forces which overcme each other. That is obvious. Hence there can be no "ALL-mighty". Further the almighty could not change irrational numbers in rational numbers and he could not go back in time to eliminate his former existence. However, if you are not capable of understanding this proof, then there is small wonder that you do not understand the rest. > > While as far as I know there very well may not be any "Almighty", no one > yet has a valid proof of that. There is no "ALL" justified, as I have shown. That is the point. Perhaps that is the reason why you write him as "Almighty" only. Regards, WM
From: mueckenh on 29 Jun 2006 05:35 Virgil schrieb: > In article <1151442095.686342.310050(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > Which digits in 0.111... = sum_{n in N} 10 ^(-n) are not indexed by > > > members of N? > > > > Best you can see it here: > > > > 1 0.1 > > 2 0.11 > > 3 0.111 > > ... ... > > n 0.111...n > > ... ... > > omega 0.111...omega > > Your "n 0,111...n" does not fit the previous pattern, as it contains > digits other than 1's, You are right, I correct that by giving the *indices* of the digits: 1 1 2 12 3 123 .... n 123...n .... w 123... w+1 123...w > and omega is not a member of the sequence > 0,1,2,... But 1,2,3,... obviously belongs to the position omega (w). > > So your example is cooked, and is meaningless. And leave out the zero. It is not a natural number. And it does not help you to explain the list above. > > >
From: imaginatorium on 29 Jun 2006 06:28
mueckenh(a)rz.fh-augsburg.de wrote: <snip-snop> > If actual infinity exists, then my mapping comes to an actual end. I may have missed some context in which this makes some sort of sense (though I doubt it, somehow)... Can I ask you then: is your concept of "actual infinity" the end of the unending? Brian Chandler http://imaginatorium.org |