Why can no one in sci.math understand my simple point?
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From: Tim Little on 28 Jun 2010 04:38 On 2010-06-27, Newberry <newberryxy(a)gmail.com> wrote: > I was replying to this: > "It implies a listing must exist, but does not provide such a > listing." > > "It", in this context is the statement "all computable reals are > countable." > > If an antidiagonal existed it would prove that there was no such > list. No, it merely implies that every such listing has an uncomputable antidiagonal, which further implies that the listing itself is an uncomputable function. - Tim
From: Tim Little on 28 Jun 2010 04:59 On 2010-06-28, Owen Jacobson <angrybaldguy(a)gmail.com> wrote: > So, here is an informal presentation of Cantor's diagonal argument > that avoids the word "list" (as well as a few other common verbal > shortcuts): > > 1. Let S be the set {0, 1}^N. > 2. For any function L from N to S, we can identify an element of S not > in the image of L. Peter is incapable of separating the usual informal phrase "we can..." from his fixed idea of "there exists a finite algorithm that can...". So, for example: 2. For any function L from N to S, there exists an element of S not in the image of L. No mention of "we can identify it" or even "given L we can find it". Those are irrelevant distractions that he will be unable to see past to the actual matter of the proof. - Tim
From: Tim Little on 28 Jun 2010 05:07 On 2010-06-28, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > This occurs in step 4. You state we can "identify an element d of S > that is on the diagonal". Unless the list is explicitly defined, you > can't "identify" the digit in position x. See Owen? I told you he wouldn't be able to ignore the informal fluff phrase "we can identify", and so fails to interpret it correctly. Mathematically, it means nothing more than "there exists". - Tim
From: WM on 28 Jun 2010 05:29 On 28 Jun., 11:07, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-28, Peter Webb <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > > This occurs in step 4. You state we can "identify an element d of S > > that is on the diagonal". Unless the list is explicitly defined, you > > can't "identify" the digit in position x. > > See Owen? I told you he wouldn't be able to ignore the informal fluff > phrase "we can identify", and so fails to interpret it correctly. > > Mathematically, it means nothing more than "there exists". And mathematically "there exists" means nothing. Regards, WM
From: Tim Little on 28 Jun 2010 05:35
On 2010-06-28, Virgil <Virgil(a)home.esc> wrote: > It does not require that any element in the listing be known, but > correctly tells what to do for any listing I think that is even a bit too informal for Peter. The phrase "tells what to do" is superfluous, all that is mathematically required is that existence of an antidiagonal sequence for each list is proven. He's going to latch onto "tells what to do" and think that it means that there is an algorithm for everything involved. Witness his confusion over the example I defined of a list where each entry was computable but the list itself (and its antidiagonal) was not. He didn't dispute that the list *existed*, but considered it cheating because he couldn't use the definition to extract actual digits of the antidiagonal - it didn't "tell him what to do" in his own special sense. - Tim |