Adriana Xenides Dead
in [Math]
|
From: |-|ercules on 8 Jun 2010 02:30 > "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote >> But I don't see how this analogy is supposed to make Cantor's theorem >> appear dubious. What about this statement? All possible digit sequences are computable to all, as in an infinite amount of, finite lengths. Herc
From: Tim Little on 8 Jun 2010 03:24 On 2010-06-08, |-|ercules <radgray123(a)yahoo.com> wrote: > What about this statement? > > All possible digit sequences are computable to all, as in an > infinite amount of, finite lengths. What about it? Apart from being an example of a mathematically ambiguous statement with at least 3 reasonable interpretations, 2 of which make it false and 1 makes it true, that is. - Tim
From: Ostap Bender on 8 Jun 2010 04:26 On Jun 8, 12:24 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-08, |-|ercules <radgray...(a)yahoo.com> wrote: > > > What about this statement? > > > All possible digit sequences are computable to all, as in an > > infinite amount of, finite lengths. > > What about it? Apart from being an example of a mathematically > ambiguous statement with at least 3 reasonable interpretations, 2 of > which make it false and 1 makes it true, that is. At least the number of reasonable interpretations is finite...
From: Graham Cooper on 8 Jun 2010 04:51 On Jun 8, 6:26 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > On Jun 8, 12:24 am, Tim Little <t...(a)little-possums.net> wrote: > > > On 2010-06-08, |-|ercules <radgray...(a)yahoo.com> wrote: > > > > What about this statement? > > > > All possible digit sequences are computable to all, as in an > > > infinite amount of, finite lengths. > > > What about it? Apart from being an example of a mathematically > > ambiguous statement with at least 3 reasonable interpretations, 2 of > > which make it false and 1 makes it true, that is. > > At least the number of reasonable interpretations is finite... I only get one interpretation which is true. It's equivalent to George Greene's wording isn't it? -Every digit sequence TO EVERY FINITE length is in the computable list -of reals. I'm not disputing it could be ambiguous, but I'm calling your bluff, what other interpretations? Herc
From: Daryl McCullough on 8 Jun 2010 07:11
|-|ercules says... > >> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote >>> But I don't see how this analogy is supposed to make Cantor's theorem >>> appear dubious. > >What about this statement? > >All possible digit sequences are computable to all, as in an infinite amount >>of, finite lengths. The correct statement is this: I. For every real number r, for every natural number n, there exists a computable real r' such that r agrees with r' in the first n decimal places. Note the logical form of this statement: forall r, forall n, exists r' ... The order of quantifiers makes a difference! If the change the order of quantifiers we get a similar-looking but false statement: II. For every real number r, there exists a computable real r', for every natural number n: r agrees with r' in the first n decimal places. This has the logical form: forall r, exists r', forall n, ... It differs from the first statement in that the order of the quantifiers has been changed. Statement I is true. Statement II is false. The claim that "not all reals are computable" is equivalent to the claim "Statement II is false". The diagonal argument proves that Statement II is false, not that Statement I is false. -- Daryl McCullough Ithaca, NY |