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Re: Simple Refutation of Cantor's Proof
Posted:
Dec 25, 2012 7:29 PM
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On Dec 26, 9:00 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <67261d2b-d57e-4174-af7b-921ac287e...@r4g2000pbi.googlegroups.com>,
> Graham Cooper <grahamcoop...@gmail.com> wrote:
>
>
> > On Dec 25, 7:22 pm, Virgil <vir...@ligriv.com> wrote:
> > > > > > [1] you change each digit ONE AT A TIME
> > > > > > 0.694...
> > > > > > but this process NEVER STOPS
>
> > > > > > [2] and you NEVER CONSTRUCT A NEW DIGIT SEQUENCE!
>
> > > > > Do you deny that f(x) = x^2 and g(x) = 2*x+3 define real functions,
> > > > > i.e., functions taking arbirary real numbers as arguments and producing
> > > > > from them appropriate real numbers as values?
>
> > > > > It you accept them as functions why balk at functions from |N to
> > > > > the set of decimal digits, interpreted as reals in [0,1]?
>
> > > > the logical manipulations do not hold on AD(x) = D(x)+1 [mod 9]
>
> > > > This is what you are really doing.
>
> > > > +----->
> > > > | 0. 542..
> > > > | 0. 983..
> > > > | 0. 143..
> > > > | 0. 543..
> > > > | ...
> > > > v
>
> > > > T(x,y) = L(x,y)+1 [mod 9]
>
> > > > +----->
> > > > | 0. 653..
> > > > | 0. 004..
> > > > | 0. 254..
> > > > | 0. 654..
> > > > | ...
> > > > v
>
> > > > This plane exists as much as your altered string.
>
> > > > It's mere naivety to define any digit string from
>
> > > > 0 . T(1,_) T(2,_) T(3,_) ...
>
> > > > where the set of free values _ biject N
> > > > and then conclude such strings are absent from L.
>
> > > > Herc
>
> > > Since that is not a rule used by anyone who knows what is needed, it is
> > > irrelevant,
>
> > > A rule that actually works on decimals, or with any base larger than 7,
> > > is to look at the nth digit of the nth element in the list and if it
> > > less than a 6 make the nth digit of the "anti-diagonal a 6 but
> > > otherwise make it a 5.
>
> > OK!
>
> > +----->
> > | 0. 542..
> > | 0. 983..
> > | 0. 143..
> > | 0. 543..
> > | ...
> > v
>
> > T(x,y) = 6 IFF L(x,y) < 6
> > T(x,y) = 5 OTHERWISE
>
> > +----->
> > | 0. 666..
> > | 0. 556..
> > | 0. 666..
> > | 0. 666..
> > | ...
> > v
>
> > This plane exists as much as your altered string.
> > It's mere naivety to define any digit string from
>
> > 0 . T(1,_) T(2,_) T(3,_) ...
>
> > where the set of free values _ biject N
> > and then conclude such strings are absent from L.
>
> > Herc
>
> Since the constructed string must differ from each string listed in L in
> a way that gives it a different value from that listed string, how can
> it possibly still be among a set of values from which it is by
> construction excluded?
>
the size of the string is not fixed.
You construct a portion, you construct another portion, and you never
stop.
Take the Transpose Plane
+----->
| 0. 666..
| 0. 556..
| 0. 666..
| 0. 666..
| ...
v
Your construction is now the DIAGONAL of that plane.
0 . T(1,1) T(2,2) T(3,3) ...
+----->
| 0. 6 ..
| 0. 5 ..
| 0. 6..
| 0.
| ...
v
so your claim is
0 . T(3,1) T(4,2) T(1,3) ...
is ALSO absent from L(x,y) right?
That's 0.666..
and your claim is
0 . T(2,1) T(4,2) T(1,3) ...
is ALSO absent from L(x,y)
That's 0.566..
So every digit you add to the missing string is arbitrary.
Herc
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