My FINAL Cantor thread!
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From: George Greene on 29 Jun 2010 00:45 On Jun 26, 5:49 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > I have no idea what you're on about. > > w is the maximum width of finite prefixes of (elements of) subsets of computable reals > such that > > there is no missing digit sequence in the finite prefixes of any considered subset If you are only considering subsets of computable reals such that no digit sequence is missing from the finite prefixes of the subset, then w does not exist. All of your "considered subsets" are going to need to have either 1) a string that is infinitely long, or 2) NO UPPER BOUND on how long the finite strings are. For your considered subsets that DON'T contain an infinite string (that contain only finite strings), THERE IS NO MAXIMUM WIDTH of finite strings in that subset! There is AN UPPER BOUND, YES, but nothing IN the set ACHIEVES the upper bound, so it IS NOT the MAXIMUM! Here's a fact about sets, lists, and other collections that you didn't know: Max(S)eS. The maximum of something IS IN it. It's the largest THING IN it. But there ARE NO infinite strings in some of the subsets you are "considering" above.
From: George Greene on 29 Jun 2010 00:46 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.
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! |