countability of real numbers? What am I missing?
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From: Saijanai on 18 May 2010 09:27 OK, all you set/number theorists, what is wrong with this binary fraction sequence. I assert it lists all real numbers [0,1] (allowing for duplicates): {0.0, 0.1}, {0.00, 0.01, 0.10, 0.11}, {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, ....
From: Tonico on 18 May 2010 09:34 On May 18, 4:27 pm, Saijanai <saija...(a)gmail.com> wrote: > OK, all you set/number theorists, what is wrong with this binary > fraction sequence. I assert it lists all real numbers [0,1] (allowing > for duplicates): > {0.0, 0.1}, > {0.00, 0.01, 0.10, 0.11}, > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, > ... Hey, wonderful! Now, just for the sake of fun, please do tell us what binary fraction from the ones you list represents...I dunno...say, the number 1/sqrt(2)? Or the numer 1/e? You know what? Forget the above: you wrote: "I assert it lists all real numbers in [0,1]...", so what about a little proof for your assertion? Tonio
From: Saijanai on 18 May 2010 09:41 On May 18, 6:34 am, Tonico <Tonic...(a)yahoo.com> wrote: > On May 18, 4:27 pm, Saijanai <saija...(a)gmail.com> wrote: > > > OK, all you set/number theorists, what is wrong with this binary > > fraction sequence. I assert it lists all real numbers [0,1] (allowing > > for duplicates): > > {0.0, 0.1}, > > {0.00, 0.01, 0.10, 0.11}, > > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111}, > > ... > > Hey, wonderful! Now, just for the sake of fun, please do tell us what > binary fraction from the ones you list represents...I dunno...say, the > number 1/sqrt(2)? Or the numer 1/e? > > You know what? Forget the above: you wrote: "I assert it lists all > real numbers in [0,1]...", so what about a little proof for your > assertion? > > Tonio The proof should be self-evident in the same way that listing the Naturals as 1, 2, 3, 4,... obviously doesn't miss any. What I'm claiming (and I'm not claiming I'm correct, only that I don't see the problem) is that the usual issues like Cantor's Diagonalization don't refute it. This is almost certainly because I don't understand Cantor's Diagonalization, but regardless, I don't see where the problem is. My own guess is that I'm conflating lexical ordering with numerical ordering but I'm not sure if that is relevant. An order is an order and a countable sequence is a countable sequence, regardless of how you arrive at them. L.
From: Torsten Hennig on 18 May 2010 05:43 > OK, all you set/number theorists, what is wrong with > this binary > fraction sequence. I assert it lists all real numbers > [0,1] (allowing > for duplicates): > {0.0, 0.1}, > {0.00, 0.01, 0.10, 0.11}, > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, > 0.111}, > ... > I only see rational numbers in your enumeration ; where are the irrational ones ? Best wishes Torsten.
From: Saijanai on 18 May 2010 09:56
On May 18, 6:43 am, Torsten Hennig <Torsten.Hen...(a)umsicht.fhg.de> wrote: > > OK, all you set/number theorists, what is wrong with > > this binary > > fraction sequence. I assert it lists all real numbers > > [0,1] (allowing > > for duplicates): > > {0.0, 0.1}, > > {0.00, 0.01, 0.10, 0.11}, > > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, > > 0.111}, > > ... > > I only see rational numbers in your enumeration ; > where are the irrational ones ? > > Best wishes > Torsten. The sequence is never-ending, so you can follow an arbitrarily long trail from level to level to create a Cauchy-Sequence that corresponds to any real, including the irrationals. My assertion is that you don't miss anything in the ordering, because the initial squence-ordering isn't numerical, just lexical. I'm not sure if that's cheating or not. I suspect it is, but I'm only auditing elementary level number theory/set theory/analysis lectures online right now and I'm no doubt missing something (or just don't understand the lectures I've already seen in the first place). L. |