My FINAL Cantor thread!
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From: Graham Cooper on 24 Jun 2010 18:50 > Consider the list of computable reals. > > Let w = the digit width of the largest set > of complete permutations > > assume w is finite > there are 10 computable copies of the > complete permutations of width w > each ending in each of digits 0..9 (at position w+1) > which generates a set larger than width w > so finite w cannot be the maximum size > > therefore w is infinite > ---- you should recognize this form of induction no maximum Natural number implies there are infinite quantity of Natural numbers IS TO no maximum digit width of all full permutation sets implies there is infinite digit width of all permutations (of computable reals) THEREFORE modifying the diagonal of the list of computable reals does not construct a new digit sequence Herc
From: Graham Cooper on 24 Jun 2010 19:09 On Jun 25, 8:50 am, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > Consider the list of computable reals. > > > > Let w = the digit width of the largest set > > of complete permutations > > > > assume w is finite > > there are 10 computable copies of the > > complete permutations of width w > > each ending in each of digits 0..9 (at position w+1) > > which generates a set larger than width w > > so finite w cannot be the maximum size > > > > therefore w is infinite > > ---- > > you should recognize this form of induction > > no maximum Natural number implies > there are infinite quantity of Natural numbers > > IS TO > > no maximum digit width of all full permutation sets implies > there is infinite digit width of all permutations I defeated myself here... There are infinite amount of digit widths. One could argue an infinite amount of finite digit widths! But I think "infinite maximum digit width" supports the claim of all possible sequences. Herc
From: George Greene on 24 Jun 2010 19:10 On Jun 24, 6:50 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > Consider the list of computable reals. > > > > Let w = the digit width of the largest set > > of complete permutations > > > > assume w is finite This is just idiotic. Every real is infinitely wide BY DEFINITION, and again, this width is, BY DEFINITION, THE SMALLEST infinity, so w is KNOWN IN ADVANCE NOT to be finite.
From: Dingo on 24 Jun 2010 19:25 On Thu, 24 Jun 2010 15:50:51 -0700 (PDT), Graham Cooper <grahamcooper7(a)gmail.com> wrote: >Herc Oh please let it be.....and if you'd said your final thread altogether that would be even more gooderer.....
From: Mike Terry on 24 Jun 2010 20:18
"Graham Cooper" <grahamcooper7(a)gmail.com> wrote in message news:df261549-3c6e-4f4d-844e-65f14cbfa8b3(a)a30g2000yqn.googlegroups.com... > > Consider the list of computable reals. > > > > Let w = the digit width of the largest set > > of complete permutations > > > > assume w is finite > > there are 10 computable copies of the > > complete permutations of width w > > each ending in each of digits 0..9 (at position w+1) > > which generates a set larger than width w > > so finite w cannot be the maximum size > > > > therefore w is infinite > > ---- or there is no largest set of complete permutations. E.g. have you considered the possibility that: 1) all 1 digit permutations are in the list 2) all 2 digit permutations are in the list ... 3) all n digit permutations are in the list ... 4) not all (countably) infinite permutations are in the list Then your w does not exist. Regards, Mike. > > you should recognize this form of induction > > no maximum Natural number implies > there are infinite quantity of Natural numbers > > IS TO > > no maximum digit width of all full permutation sets implies > there is infinite digit width of all permutations > (of computable reals) > > THEREFORE > > modifying the diagonal of the list of computable reals > does not construct a new digit sequence > > Herc |