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: George Greene on 24 Jun 2010 19:09 On Jun 24, 6:50 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > Consider the list of computable reals. NO. You may NOT consider that. In the first place, THAT list is itself NOT computable. In the second place, EVERYthing you are WANTING to claim about that list can be JUST as correctly claimed about the list of all FINITE bit-strings (which IS computable). > > Let w = the digit width of the largest set of complete permutations No. w may NOT equal that, since THERE IS NO SUCH THING as a "permutation" for purposes of this question. What you are TRYING TO CALL a "permutation" IS JUST A SEQUENCE -- IS JUST A LIST, and every one of them IS DIFFERENT AND SEPARATE IN ITS OWN RIGHT. THERE IS NO important sense in which some of them are permutations of others, and even if there were, YOU WOULDN'T KNOW WHICH to call the "base" or identity permutations and which to call "permuted". A sequence is a sequence. A list is a list. The fact that it is a permuation of its other permutations IS NOT relevant! More to the point, if you are talking about reals, ABSOLUTELY ALL OF THEM, ALL THE TIME, are of width w IF they are going to be expressed as digit-strings. So there is no point in talking about ANY OTHER width, or asking about "the largest set": EVEN THE SMALLEST non-empty set, the one with ONE real, IS OF WIDTH W. No matter how many more reals you may add to get a bigger set, SINCE THEY WILL ALL, ALWAYS, BE OF WIDTH w, w IS THE ONLY width that will ever be relevant to this question. And w is A CONSTANT. It is NOT a variable or a statistic or something that you might need to compute. It is KNOWN EXACTLY, IN ADVANCE.
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..... |