THE CANTOR ARGUMENT SO FAR
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From: Rupert on 20 Jun 2010 01:12 On Jun 20, 8:21 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > ------------------------SCI.MATH----------------------------- > > Take any list of reals > > 123 > 456 > 789 > > Diag = 159 > AntiDiag = 260 > > It's a NEW DIGIT SEQUENCE and it works on EVERY LIST. > > ---------------------------HERC------------------------------ > > defn(herc_cant_3) > The list of computable reals contains every digit (in order) of all possible infinite sequences. > > ..as a result of containing ALL (infinitely many) finite prefixes. > > THEREFORE YOU CANNOT CONSTRUCT A NEW DIGIT SEQUENCE > > --------------------------SCI.MATH-------------------------- > > BUT: > > 0.0 > 0.1 > 0.2 > ... > 0.01 > 0.02 > 0.03 > ... > 0.99 > 0.101 > 0.102 > ... > > ALSO contains every finite prefix > > AND 0.111... is not on that list. > > THEREFORE ANTI-DIAG STILL *IS* A NEW DIGIT SEQUENCE. > > -----------------------------HERC------------------------------ > > A correction to a correction does not prove the original assertion. > > You STILL have not come up with a NEW DIGIT SEQUENCE. > > You use the term NEW DIGIT SEQUENCE for the finite example 260 > then you BAIT AND SWITCH and call it NEW NUMBER because > An AD(n) =/= L(n,n). > > Is it a *NEW DIGIT SEQUENCE* or not? > > Herc 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.
From: |-|ercules on 20 Jun 2010 02:02 "Rupert" <rupertmccallum(a)yahoo.com> wrote > On Jun 20, 8:21 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> ------------------------SCI.MATH----------------------------- >> >> Take any list of reals >> >> 123 >> 456 >> 789 >> >> Diag = 159 >> AntiDiag = 260 >> >> It's a NEW DIGIT SEQUENCE and it works on EVERY LIST. >> >> ---------------------------HERC------------------------------ >> >> defn(herc_cant_3) >> The list of computable reals contains every digit (in order) of all possible infinite sequences. >> >> ..as a result of containing ALL (infinitely many) finite prefixes. >> >> THEREFORE YOU CANNOT CONSTRUCT A NEW DIGIT SEQUENCE >> >> --------------------------SCI.MATH-------------------------- >> >> BUT: >> >> 0.0 >> 0.1 >> 0.2 >> ... >> 0.01 >> 0.02 >> 0.03 >> ... >> 0.99 >> 0.101 >> 0.102 >> ... >> >> ALSO contains every finite prefix >> >> AND 0.111... is not on that list. >> >> THEREFORE ANTI-DIAG STILL *IS* A NEW DIGIT SEQUENCE. >> >> -----------------------------HERC------------------------------ >> >> A correction to a correction does not prove the original assertion. >> >> You STILL have not come up with a NEW DIGIT SEQUENCE. >> >> You use the term NEW DIGIT SEQUENCE for the finite example 260 >> then you BAIT AND SWITCH and call it NEW NUMBER because >> An AD(n) =/= L(n,n). >> >> Is it a *NEW DIGIT SEQUENCE* or not? >> >> Herc > > 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. Hypothesis: a real number contains a finite sequence that is not computable. Contradiction Therefore: all digits of every real are contained in the list of computable reals. _________________________________________________________________ This may not IMPLY that all infinite digit sequences are computable, but it trivially defeats this argument: 123 456 789 Diag = 159 AntiDiag = 260 A new digit sequence can be found on all real lists. Herc
From: Sylvia Else on 20 Jun 2010 02:50 On 20/06/2010 12:42 PM, |-|ercules wrote: > "George Greene" <greeneg(a)email.unc.edu> wrote >>> Is it a *NEW DIGIT SEQUENCE* or not? >> >> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was >> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! > > > But you keep saying the anti-diagonal is NEW and ignoring me when > I say it's not a new digit sequence. > > Then you repeat Cantor's proof again and again that it's NEW. > > You use terms NEW and NOT ON THE LIST, but evade me when I challenge > whether it contains any new digit sequence. And you just ignore the point that it must be new because it demonstrably isn't in the list. Sylvia.
From: |-|ercules on 20 Jun 2010 02:59 "George Greene" <greeneg(a)email.unc.edu> wrote >> But I still maintain all possible variations of digit sequences are present >> up to infinite width on the list. > > You DO NOT maintain that. > A number is either on the list or it isn't. > All FINITE PREFIXES of a number being on the list IS NOT THE SAME > THING AS > THE NUMBER being on the list! I still maintain all possible variations of digit sequences are present up to infinite width on the list. Herc
From: |-|ercules on 20 Jun 2010 03:03
"Sylvia Else" <sylvia(a)not.here.invalid> wrote ... > On 20/06/2010 12:42 PM, |-|ercules wrote: >> "George Greene" <greeneg(a)email.unc.edu> wrote >>>> Is it a *NEW DIGIT SEQUENCE* or not? >>> >>> YES, DUMBASS, IT IS A NEW DIGIT-SEQUENCE because it was >>> NOT ON THE LIST of (allegedly "all") THE OLD digit-sequences! >> >> >> But you keep saying the anti-diagonal is NEW and ignoring me when >> I say it's not a new digit sequence. >> >> Then you repeat Cantor's proof again and again that it's NEW. >> >> You use terms NEW and NOT ON THE LIST, but evade me when I challenge >> whether it contains any new digit sequence. > > And you just ignore the point that it must be new because it > demonstrably isn't in the list. > > Sylvia. All you demonstrated was An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) You don't like axioms stating a fact, but you use a definition as a proof. Herc |