Why can no one in sci.math understand my simple point?
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From: Charlie-Boo on 27 Jun 2010 04:13 On Jun 26, 3:45 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <b5f0d60a-f8d6-4e40-a180-a9dbbfd28...(a)u26g2000yqu.googlegroups.com>, > > Charlie-Boo <shymath...(a)gmail.com> wrote: > > It is trivial to calculate in binary (using only 0 and 1) and output > > any desired string surrounded by 2 whenever we want, then immediately > > erase one of the 2's. Thus every real number will be defined by some > > TM. > > How does one calculate one of those many reals that one cannot > calculate, We cannot calculate them. (From a paper written at the University of Duh.) C-B > the inaccessible ones? > Note that most reals are inaccessible, uncountably many of them, with > only countably many being accessible.
From: Charlie-Boo on 27 Jun 2010 04:27 On Jun 26, 8:48 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > >> Example 1: The TM that never halts and never changes the tape does not > >> represent a computable real. > > > It depends on the system of representation. An empty tape typically > > represents 0. > > If the machine does not halt, you can't say that the tape is empty for > certain. Likewise if it does halt - until it halts. But who said that you have to say anything? > The usual definitions of computable real require that the machine accept > input n, halts and outputs the first n digits, or something like that. Well, you could do that too if you want, but in that case not all TM define a number. You are simply choosing a sparse representation instead of a dense one. You know how when they're defining a system they either have the set of wffs be represented by a sequential (dense) number or a number where only certain (r.e. set) numbers represent a wff? You're doing it the second way. If you do it the first way you can have a total map from TM onto the computable real numbers. > >> Example 2: The TM that repeatedly changes the value in one cell, never > >> halting, does not represent a computable real. > > > The sequence of values put into a given cell defines the decimal > > expansion of a real number. Every machine must distinguish between > > scratch values and actual output. Having a certain cell represent > > output is also common. > > > Google "Turing Machines". > > So, what real number do you think example 2 computes? And what > convention of computable real number do you have in mind? It represents the real number whose expansion is the sequence of non- blank values set into that cell. > Show me a reference or web page, rather than telling me to google Turing > Machines. > > In other words, stop bluffing. I'm not bluffing - you really can Google "Turing Machines". Or better yet, Google "Quine Atom" and click I'm Feeling Lucky. (Google's team of experts carefully ranks all sites of a scholarly nature.) C-B > -- > Jesse F. Hughes > "It's easy folks. Just talk about my approach to your favorite > mathematician. If they can't be interested in it, they've > demonstrated a lack of mathematical skill." -- James Harris
From: Charlie-Boo on 27 Jun 2010 04:29 On Jun 26, 11:19 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-26, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > It is trivial to calculate in binary (using only 0 and 1) and output > > any desired string surrounded by 2 whenever we want > > You use the singular here, by which your statement is correct. Any > single finite string can be so produced. What's more, any finite > sequence of finite strings can be produced. > > However, you need an infinite sequence of finite strings to define > most real numbers, and there are not enough finite algorithms to > produce them all. Why not? We're talking about computable reals. C-B > - Tim
From: Charlie-Boo on 27 Jun 2010 04:33 On Jun 26, 11:33 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-27, Jesse F. Hughes <je...(a)phiwumbda.org> wrote: > > > The usual definitions of computable real require that the machine > > accept input n, halts and outputs the first n digits, or something > > like that. > > There are a number of definitions, all of which end up being provably > equivalent. The definition CB has so far provided is a particularly > bizarre one, but does still appear to be equivalent. What is so particularly bizarre about it? However, I have to admit that once a Boston TV station said I was something like "peculiar" - I forget the word. (I'll post the video on my home page some day.) C-B > > - Tim
From: Charlie-Boo on 27 Jun 2010 04:35
On Jun 27, 12:08 am, Virgil <Vir...(a)home.esc> wrote: > In article <slrni2dgpo.jrj....(a)soprano.little-possums.net>, > Tim Little <t...(a)little-possums.net> wrote: > > > On 2010-06-26, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > It is trivial to calculate in binary (using only 0 and 1) and output > > > any desired string surrounded by 2 whenever we want > > > You use the singular here, by which your statement is correct. Any > > single finite string can be so produced. What's more, any finite > > sequence of finite strings can be produced. > > > However, you need an infinite sequence of finite strings to define > > most real numbers, and there are not enough finite algorithms to > > produce them all. > > It is well known that for 'most' reals no such algorithm can exist, the > non-computable transcendentals. That has no bearing on this discussion. (They need not be transcendental.) C-B |