My FINAL Cantor thread!
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From: |-|ercules on 29 Jun 2010 02:04 "George Greene" <greeneg(a)email.unc.edu> wrote > On Jun 26, 5:49 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> w is the maximum width of finite prefixes of (elements of) subsets > > FORGET subsets. THERE ARE NO relevant subsets here. > THE ONLY thing that is relevant is THE LIST OF ALL FINITE > digit-sequences. > This list is countable and computable, and every finite prefix OF > EVERY > real (computable, non-computable, yellow, purple, or polka-dotted) IS > ON it. The proof is not induction on finite sequences, it is induction on finite prefixes. Learn the difference! Herc
From: George Greene on 29 Jun 2010 15:26 On Jun 29, 2:04 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > The proof is not induction on finite sequences, it is induction on finite prefixes. > Learn the difference! You DON'T KNOW jack about induction. Proving something for all finite prefixes IS EXACTLY THE SAME as proving it for all finite sequences: EVERY finite prefix IS a finite sequence, and EVERY finite sequence IS a finite prefix (of the string consisting of itself concatenated with ANOTHER 0).
From: George Greene on 29 Jun 2010 15:28 On Jun 27, 7:16 am, "Mike Terry" <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > That's not the way you defined w at the start of the thread. THAT is NOT the issue! The issue IS that this definition IS INCOHERENT! Was the one at the start any better???? And I still insist you can't tolerate "w" as the letter for this, because w IS ACTUALLY THE RIGHT width ("w" is ascii for lower-case-greek omega, WHICH IS RIGHT). But w as he is trying to define it is too meaningless to be wrong -- you cannot define "w" OR ANYTHING ELSE as the maximum of a series THAT DOES NOT HAVE a maximum!
From: |-|ercules on 29 Jun 2010 18:36 "George Greene" <greeneg(a)email.unc.edu> wrote .. > On Jun 29, 2:04 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> The proof is not induction on finite sequences, it is induction on finite prefixes. >> Learn the difference! > > You DON'T KNOW jack about induction. > Proving something for all finite prefixes IS EXACTLY THE SAME > as proving it for all finite sequences: EVERY finite prefix IS a > finite sequence, > and EVERY finite sequence IS a finite prefix (of the string > consisting of itself > concatenated with ANOTHER 0). Let's call that a default_finite_prefix. Then every finite sequence being a default_finite_prefix does not make them equivalent. You prove a property for increasing different objects. I sample larger and larger sizes of the one object. Different style of proof! I prove, by induction, the (anti-transfiniteness) property holds for all digit widths (all digits) You prove, by induction, that all (finite) sizes of prefixes the (pro-transfiniteness) property holds. Herc
From: |-|ercules on 29 Jun 2010 19:53
"George Greene" <greeneg(a)email.unc.edu> wrote > On Jun 27, 7:16 am, "Mike Terry" > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: >> That's not the way you defined w at the start of the thread. > > THAT is NOT the issue! Correct! > The issue IS that this definition IS > INCOHERENT! Wrong on both counts This definition is entirely clear, and the issue is whether Mike and others can review the proof with the clearer definition. > Was the one at the start any better???? No. That's the issue. > > And I still insist you can't tolerate "w" as the letter for this, Like I said, use another letter. A million variable declarations using w are written by mathematicians and students every day, none of them are referring to infinity, 99% don't even realize the connotation of the letter. w is for width. You said yourself it's a good coincidence. I was being clever since the proof shows w IS infinity. A double meaning lost on you. > because w IS ACTUALLY THE RIGHT width ("w" is ascii for > lower-case-greek omega, WHICH IS RIGHT). But w as he is trying to > define > it is too meaningless to be wrong -- you cannot define "w" OR ANYTHING > ELSE > as the maximum of a series THAT DOES NOT HAVE a maximum! So you can't define a list of all reals, a function that determines if a program halts.. For each subset of computable reals, there exists a maximum digit length that that subset doesn't miss a possible sequence of digits. w is the maximum of those maximums. GOT IT! Herc |