My FINAL Cantor thread!
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From: George Greene on 6 Jul 2010 12:38 On Jul 5, 10:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "George Greene" <gree...(a)email.unc.edu> wrote > > > On Jul 5, 7:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >> "George Greene" <gree...(a)email.unc.edu> wrote > > >> > On Jul 5, 8:36 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >> >> It's the 'covered sequences' within the infinite sequences > >> >> that approaches infinity. > > > You still have to DEFINE COVERED, DUMBASS. > > You made a post last week using some word to mean what I mean, but I shan't bother looking for it. > > > And it DOES NOT "approach infinity". It just STAYS FINITE ALL the > > time. > > Binary example > > 00000000... > 01111111... > 01011111... > 01000000... > 01010101... > 00111111... > 11111111... > 11000000... > 11011111... > 10000000... > 10111111... > 11000000... > > The length of all (initial) possible digit sequences within the set is 3. This is A LIST, NOT a set. This HAS an order (sets DON'T). It is stupid for this list to be FINITE. What you NEED AND WANT is THE INFINITE list OF ALL "initial possible digit sequences". You are finite THE WRONG WAY. You are finite, here, VERTICALLY. What you NEED is INFINITELY long vertically but finite HORIZONTALLY. You DON'T NEED ANY infinitely long reals in this list, COMPUTABLE OR OTHERWISE! ALL YOU NEED (to cover all the sequences) is the list OF ALL *finite* sequences! And in THAT case, you would not even need TO WORRY about COVERING anything because every finite initial sequence WOULD BE*ON*THE LIST, AS AN ELEMENT! > So you are saying this length does not approach infinity as the length of the computable reals list approaches infinity? No, I'm saying that the length of the computable reals list DOES NOT AND CANNOT *approach* anything BECAUSE IT *IS*ALWAYS* A CONSTANT, namely w (the smallest infinity). What you were actually trying to do was take initial segments (VERTICALLY) of the computable reals list, and ask whether a finite initial segment of the computable reals INCLUDED, AS AN ELEMENT -- NOT "covered" -- an element that started with (had as a prefix) every one of the 10^n possible width-n digit-sequences. This is a stupid question. If you want a list that has all the width-n prefixes on it THEN YOU JUST WRITE A LIST OF THEM. IT DOES NOT MATTER what "the list of computable reals" says about the parts of the reals that are wider than that!
From: George Greene on 6 Jul 2010 12:52
On Jul 5, 10:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > Binary example > > 00000000... > 01111111... > 01011111... > 01000000... > 01010101... > 00111111... > 11111111... > 11000000... > 11011111... > 10000000... > 10111111... > 11000000... > > The length of all (initial) possible digit sequences within the set is 3. Until you say "covered means ...." and fill in the blank with something mathematically meaningful, you are just not going to be allowed to talk. The fact that you can't speak English well enough to do this basically means you should've been censored a long time ago. Right now, you have switched to using "covered" in a different way. If you have programming experience then you know that variables or function-arguments often have TYPES or shapes. You cannot apply a function or procedure or anything else to arguments of THE WRONG type. If you are going to use "covered" then you have to be clear in YOUR OWN mind about what TYPES of things CAN legally get "covered". In the above, you are (newly) using "covered" to mean something that happens TO A FINITE SET of all "permutations" of A CONSTANT LENGTH of digit-strings. There are 2^3=8 bit-strings of length 3, and BECAUSE there are only 8 of them, ANY OLD 8-element list COULD manage to cover them, IF you wanted it to (if you put the right things at the beginning of it). YOU DO NOT NEED a LONG list to do this. Obviously, if you want to cover all 10^n sequences FOR ALL n, THEN you will need an infinite list since there are infintely many n's. But you still don't need anything infinitely WIDE to do that: you could "cover" all these JUST BY LISTING them. You would not even NEED the CONCEPT of "covering" AT ALL! YOU ONLY NEED to worry about "covering" when you are trying to cover something INFINITELY WIDE, LIKE Pi! Your talking about "covering" A FINITE set IS JUST STUPID! If the set is FINITE then you can just put EVERY element of it ON the list, without even WORRYING about what the list COVERS! |