THE CANTOR ARGUMENT SO FAR
in [Logic]
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From: George Greene on 20 Jun 2010 10:35 On Jun 20, 3:03 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > All you demonstrated was > > An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) This is NOT ALL we demonstrated. BECAUSE we demonstrated this, we HAVE THEREFORE demonstrated that AD(L) IS NOT ON the list L! > You don't like axioms stating a fact, YOU ARE A DAMN LIAR AND A DAMN FOOL. WE DO SO TOO like "Axioms stating a fact"! That's what an axiom IS! It is A FACT about the little abstract/mathematical sub-universe that WE are TRYING to talk about (we call this "the universe of discourse", for this PARTICULAR discourse -- if you want to have a DIFFERENT discourse about OTHER things, FINE). > but you use a definition as a proof. Dipshit: YOU DON'T KNOW THE DIFFERENCE between a definition and a proof. JUST WHAT definition did we use? And we don't use anything ELSE AS a proof: we use A PROOF as a proof! Cantor's Theorem HAS A PROOF (that's what MAKES it a theorem) and YOU HAVE SEEN this proof! And you have NOT shown any errors in it! Yet you persist! > > Herc
From: George Greene on 20 Jun 2010 10:38 On Jun 20, 2:59 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > I still maintain all possible variations of digit sequences are present > up to infinite width on the list. NO, YOU DON'T. NObody maintains that ANYthing "is present up to" ANYthing ON A LIST, BECAUSE, on a list, THE ONLY question is whether something IS AN ELEMENT OF the list, IS ON the list, OR NOT. EVEN IF YOU HAD a coherent definition of "being present up to infinite width", IT WOULDN'T MATTER, because things so present STILL WOULD *NOT*BE*ON* the list! More to the point, when you say "up to", does that INCLUDE OR NOT INCLUDE the VERY LAST thing?? Please be aware that the list of natural numbers DOES NOT HAVE a last element. You cannot go "up to" INFINITY by going through ALL the natural numbers because EVEN THAT never takes you TO infinity!! Do you, or do you not, recognize a DIFFERENCE between going "up to" Boston and going TO Boston????
From: George Greene on 20 Jun 2010 10:42 On Jun 20, 4:49 am, Sylvia Else <syl...(a)not.here.invalid> wrote: > In what sense could a number be said to be in a list if it doesn't match > any element of the list? In the sense of being present up to infinite width "on" the list. He can (or could, if he weren't a dipshit) say that "a digit-string is present up to finite width m on the list if the first m digits of the string and the first m digits of an element THAT IS ON the list ALL MATCH". We could all agree that that MEANS that. But he will then say that if this holds for all finite m (which of course it does, if he would just agree to use the list of all finite strings), then it holds "up TO infinite width", which is, obviously, a VERY problematic locution, since it in fact DOES NOT hold for the (unique) INFINITE width for any INDIVIDUAL ELEMENT of the list. Nevertheless, he could still claim it held for the list as a whole. Which, if he wanted to use words this way, IT WOULD. BUT THIS STILL WOULD NOT IMPLY that the number was not NEW! The fact that the number would STILL NOT BE ON the list would imply that it WAS new. But he seems to think that just because he can misuse English, that somehow makes the number old.
From: George Greene on 20 Jun 2010 10:49 On Jun 20, 1:12 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > Let L be a countably infinite list of countably infinite sequences of > decimal digits. Cantor's diagonal construction shows how to construct > a sequence of decimal digits which is not in L. It is not in L, > because, given any sequence which is in L, we can find a position for > which the sequence in L differs from the sequence constructed by > Cantor's diagonal construction. You are late to the point of stupidity. Why do you think Herc is going to do any better with THAT than he did with anything else? Herc KNOWS ALL THAT already. Herc is making THE OPPOSITE point that FOR ALL m, there is an element THAT IS ON the list that MATCHES NOT ONLY the anti-diagonal, BUT ANY AND EVERY POSSIBLE real, for the first m places. He thinks (wrongly) that this means something. He thinks (wrongly) that this implies something about what's new and what's old. He thinks this means that any and every possible real is therefore "present up to infinite width" in some nebulous sense requiring a new preposition all its own, such as "within" the list. The problem really boils DOWN to what he means by "UP to". If you said "this number is present up to width 5 on this list" then Most people would think that that would mean that position 1-5 were matched by some number on the list. The fact that you can do this up to arbitrarily high FINITE widths IS NOT THE SAME AS being able to do it "up to infinite" width, but Herc (wrongly) THINKS THAT IT IS OK TO TALK THAT WAY. You are NOT DEALING with a mathematical issue here! You are dealing with an abuse of NATURAL language!!
From: George Greene on 20 Jun 2010 10:54
On Jun 20, 5:42 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > What sense is a number that's only definition is to not be on a list? THAT SENTENCE IS NOT IN ENGLISH, DIPSHIT. If you can rephrase that question GRAMMATICALLY, correctly, THEN somebody will answer it. The definition of the anti-diagonal MAKES PERFECT sense, FOR ANY square list. IT IS ALWAYS WELL defined. It simply IS NOT THE CASE that the anti- diagonal's "only definition is not to be on a list". It is simply a number computed from the list. From any list, you could create a new number by picking 1 column from every row. We just picked the SIMPLEST way to do that, picking the column that MATCHED the row, to get the diagonal. Once you have the diagonal of the list, THAT IS A REAL NUMBER. There is nothing strange or unusual about its definition. And once you have ANY digit-sequence, you can just take 9-d for every digit in it, i.e., you can subtract it from ..999999...... For ANY real r between 0 and 1, 1-r IS WELL defined! THAT IS ALL we are doing here! Each step of the definition of the number makes PERFECT sense, EVEN TO YOU. So stop pretending otherwise. |