> > > > You change the DIAGONAL to the ANTI-DIAGONAL and no existing real can
> > > > be left on the list!

> > > Tell me how to produce the antidiagonal of an infinite list.

> > One method is to input a stream of digits and interpret them as a
> > transpose sort of the list.

> Wait a second.
> I'm accepting a minor repurposing of the word as defined athttp://en.wiktionary.org/wiki/antidiagonal
> What do you mean by the antidiagonal of an infinite list of
> real numbers with infinite digits after the decimal point?
> Where do you find the upper right or bottom left corner of
> an infinitely large matrix?  I get the diagonal because it
> defines the first digit after the decimal point in the first
> member of the set, the second digit after the decimal point in
> the second member of the set, the nth digit after the decimal
> point in the nth member of the set.

> The part I cut dealt with finite sets and an arbitrary cutoff
> of digits after the decimal point.  The routine you provided
> doesn't apply to Cantor's diagonalization let alone refute it.

> Here's a version of Cantor's argument.
> Given any ordered set of unique reals on the interval (0,1)
> represented base ten where the order is identified via a map
> to the natural numbers in their normal order, then...
> given the identification of the elements using the index function
> R=f(x) where x is index into f and R is the value the real number
> at that index... then the real number developed by summing
> case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> for all x won't be in the set.  The reason is it differs from
> f(x) in at least the xth digit after the decimal.

> If it differs from all reals on the list, then by what Epsilon>0
> is the difference?

> > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > Here's a version of Cantor's argument.
> > > Given any ordered set of unique reals on the interval (0,1)
> > > represented base ten where the order is identified via a map
> > > to the natural numbers in their normal order, then...
> > > given the identification of the elements using the index function
> > > R=f(x) where x is index into f and R is the value the real number
> > > at that index... then the real number developed by summing
> > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > for all x won't be in the set.  The reason is it differs from
> > > f(x) in at least the xth digit after the decimal.

> > If it differs from all reals on the list, then by what Epsilon>0
> > is the difference?

> There is no Epsilon of course.  Compare: sqrt(2) differs from all
> rationals
> but there is no Epsilon>0 by which it differs from all rationals

> > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > Here's a version of Cantor's argument.
> > > > Given any ordered set of unique reals on the interval (0,1)
> > > > represented base ten where the order is identified via a map
> > > > to the natural numbers in their normal order, then...
> > > > given the identification of the elements using the index function
> > > > R=f(x) where x is index into f and R is the value the real number
> > > > at that index... then the real number developed by summing
> > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > for all x won't be in the set.  The reason is it differs from
> > > > f(x) in at least the xth digit after the decimal.

> > > If it differs from all reals on the list, then by what Epsilon>0
> > > is the difference?

> > There is no Epsilon of course.  Compare: sqrt(2) differs from all
> > rationals
> > but there is no Epsilon>0 by which it differs from all rationals

> A powerful analogy that turned the entire maths world into fools.

> Herc

> --
> "Nothing supernatural there!" ~ Bob Cwww.chatzombie.com/BUDDHA-CLOUD.gif

> > On Nov 10, 9:56 am, William Hughes <wpihug...@gmail.com> wrote:

> > > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > > Here's a version of Cantor's argument.
> > > > > Given any ordered set of unique reals on the interval (0,1)
> > > > > represented base ten where the order is identified via a map
> > > > > to the natural numbers in their normal order, then...
> > > > > given the identification of the elements using the index function
> > > > > R=f(x) where x is index into f and R is the value the real number
> > > > > at that index... then the real number developed by summing
> > > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > > for all x won't be in the set.  The reason is it differs from
> > > > > f(x) in at least the xth digit after the decimal.

> > > > If it differs from all reals on the list, then by what Epsilon>0
> > > > is the difference?

> > > There is no Epsilon of course.  Compare: sqrt(2) differs from all
> > > rationals
> > > but there is no Epsilon>0 by which it differs from all rationals

> > A powerful analogy that turned the entire maths world into fools.

> Not an analogy, a counterexample

>     proposition, if x differs from every element of S. then there is
> an
>                  Epsilon>0 such that x differs by at least epsilon

>     counterexample:  x=sqrt(2), S the irrationals

> > On Nov 10, 9:56 am, William Hughes <wpihug...@gmail.com> wrote:

> > > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > > Here's a version of Cantor's argument.
> > > > > Given any ordered set of unique reals on the interval (0,1)
> > > > > represented base ten where the order is identified via a map
> > > > > to the natural numbers in their normal order, then...
> > > > > given the identification of the elements using the index function
> > > > > R=f(x) where x is index into f and R is the value the real number
> > > > > at that index... then the real number developed by summing
> > > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > > for all x won't be in the set.  The reason is it differs from
> > > > > f(x) in at least the xth digit after the decimal.

> > > > If it differs from all reals on the list, then by what Epsilon>0
> > > > is the difference?

> > > There is no Epsilon of course.  Compare: sqrt(2) differs from all
> > > rationals
> > > but there is no Epsilon>0 by which it differs from all rationals

> > A powerful analogy that turned the entire maths world into fools.

> Not an analogy, a counterexample

>     proposition, if x differs from every element of S. then there is
> an
>                  Epsilon>0 such that x differs by at least epsilon

>     counterexample:  x=sqrt(2), S the irrationals

> > > On Nov 10, 9:56 am, William Hughes <wpihug...@gmail.com> wrote:

> > > > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > > > Here's a version of Cantor's argument.
> > > > > > Given any ordered set of unique reals on the interval (0,1)
> > > > > > represented base ten where the order is identified via a map
> > > > > > to the natural numbers in their normal order, then...
> > > > > > given the identification of the elements using the index function
> > > > > > R=f(x) where x is index into f and R is the value the real number
> > > > > > at that index... then the real number developed by summing
> > > > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > > > for all x won't be in the set. The reason is it differs from
> > > > > > f(x) in at least the xth digit after the decimal.

> > > > > If it differs from all reals on the list, then by what Epsilon>0
> > > > > is the difference?

> > > > There is no Epsilon of course. Compare: sqrt(2) differs from all
> > > > rationals
> > > > but there is no Epsilon>0 by which it differs from all rationals

> > > A powerful analogy that turned the entire maths world into fools.

> > Not an analogy, a counterexample

> > proposition, if x differs from every element of S. then there is
> > an
> > Epsilon>0 such that x differs by at least epsilon

> > counterexample: x=sqrt(2), S the irrationals

> And <1 2 3...>  is different to all of

> <1 2 1 2 1 2...>
> <3 3 3 3 3 3...>
> <9 8 7 9 8 7...>

> for a specific reason
> =================

> <9 9 8 6 2 5 8 3 4...>

> is not different to all of

> <4 2 0 5 3 4 3 4 2..>
> <2 4 3 5 6 3 4 4 3..>
> <0 3 3 3 0 4 4 5 5..>
> ...
> AND SO ON..

> just because you can formulate a pinpoint intersection of digit
> inequality.

> MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>

> It's total nonsense if you work with infinite )sets_ of reals.

> > On Nov 10, 3:33 pm, William Hughes <wpihug...@gmail.com> wrote:
> > > On Nov 10, 12:39 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > On Nov 10, 9:56 am, William Hughes <wpihug...@gmail.com> wrote:

> > > > > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > > > > Here's a version of Cantor's argument.
> > > > > > > Given any ordered set of unique reals on the interval (0,1)
> > > > > > > represented base ten where the order is identified via a map
> > > > > > > to the natural numbers in their normal order, then...
> > > > > > > given the identification of the elements using the index function
> > > > > > > R=f(x) where x is index into f and R is the value the real number
> > > > > > > at that index... then the real number developed by summing
> > > > > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > > > > for all x won't be in the set.  The reason is it differs from
> > > > > > > f(x) in at least the xth digit after the decimal.

> > > > > > If it differs from all reals on the list, then by what Epsilon>0
> > > > > > is the difference?

> > > > > There is no Epsilon of course.  Compare: sqrt(2) differs from all
> > > > > rationals
> > > > > but there is no Epsilon>0 by which it differs from all rationals

> > > > A powerful analogy that turned the entire maths world into fools.

> > > Not an analogy, a counterexample

> > >     proposition, if x differs from every element of S. then there is
> > > an
> > >                  Epsilon>0 such that x differs by at least epsilon

> > >     counterexample:  x=sqrt(2), S the irrationals

> > And <1 2 3...>  is different to all of

> > <1 2 1 2 1 2...>
> > <3 3 3 3 3 3...>
> > <9 8 7 9 8 7...>

> > for a specific reason
> > =================

> > <9 9 8 6 2 5 8 3 4...>

> > is not different to all of

> > <4 2 0 5 3 4 3 4 2..>
> > <2 4 3 5 6 3 4 4 3..>
> > <0 3 3 3 0 4 4 5 5..>
> > ...
> > AND SO ON..

> > just because you can formulate a pinpoint intersection of digit
> > inequality.

> > MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>

> > It's total nonsense if you work with infinite )sets_ of reals.

> On the contrary, it works quite well for countably infinite sets of
> reals in lists.

> > > On Nov 10, 3:33 pm, William Hughes <wpihug...@gmail.com> wrote:
> > > > On Nov 10, 12:39 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > > On Nov 10, 9:56 am, William Hughes <wpihug...@gmail.com> wrote:

> > > > > > On Nov 9, 5:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > > > > > On Nov 9, 7:18 pm, forbisga...@gmail.com wrote:
> > > > > > > > Here's a version of Cantor's argument.
> > > > > > > > Given any ordered set of unique reals on the interval (0,1)
> > > > > > > > represented base ten where the order is identified via a map
> > > > > > > > to the natural numbers in their normal order, then...
> > > > > > > > given the identification of the elements using the index
> > > > > > > > function
> > > > > > > > R=f(x) where x is index into f and R is the value the real
> > > > > > > > number
> > > > > > > > at that index... then the real number developed by summing
> > > > > > > > case (floor(f(x) * 10^(x+1)) mod 10 ) = 0 then 7 * 0.1^n
> > > > > > > > else ((floor(f(x) * 10^(x+1)) mod 10) - 1) * 0.1^n
> > > > > > > > for all x won't be in the set.  The reason is it differs from
> > > > > > > > f(x) in at least the xth digit after the decimal.

> > > > > > > If it differs from all reals on the list, then by what Epsilon>0
> > > > > > > is the difference?

> > > > > > There is no Epsilon of course.  Compare: sqrt(2) differs from all
> > > > > > rationals
> > > > > > but there is no Epsilon>0 by which it differs from all rationals

> > > > > A powerful analogy that turned the entire maths world into fools.

> > > > Not an analogy, a counterexample

> > > >     proposition, if x differs from every element of S. then there is
> > > > an
> > > >                  Epsilon>0 such that x differs by at least epsilon

> > > >     counterexample:  x=sqrt(2), S the irrationals

> > > And <1 2 3...>  is different to all of

> > > <1 2 1 2 1 2...>
> > > <3 3 3 3 3 3...>
> > > <9 8 7 9 8 7...>

> > > for a specific reason
> > > =================

> > > <9 9 8 6 2 5 8 3 4...>

> > > is not different to all of

> > > <4 2 0 5 3 4 3 4 2..>
> > > <2 4 3 5 6 3 4 4 3..>
> > > <0 3 3 3 0 4 4 5 5..>
> > > ...
> > > AND SO ON..

> > > just because you can formulate a pinpoint intersection of digit
> > > inequality.

> > > MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>

> > > It's total nonsense if you work with infinite )sets_ of reals.

> > On the contrary, it works quite well for countably infinite sets of
> > reals in lists.

> You equate LIST with COUNTABLE SET
> hence you missed the point completely.

> There is no diagonal of an infinite set.

> It's total nonsense if you work with infinite )sets_ of reals.

> <snip>

> > MODIFY-ALL-OF-SEQUENCE <4 4 3 ... > = <9 9 8 ...>

> > It's total nonsense if you work with infinite )sets_ of reals.

> sqrt(2) an infinite sequence of digits you can define.
> ANTIDIAG(L) an infinite sequence of digits you cannot define.

> Lists ARE countable sets, though not all sets need to be lists.

> > Lists ARE countable sets, though not all sets need to be lists.

> This PROVES you MISSED THE POINT!

> Countable SETS are NOT LISTS.

> This set has no diagonal.

> 0.102  0.434  0.543
> 0.245  0.545  0.544
> 0.095  0.111  0.999

> Herc

> > > > You equate LIST with COUNTABLE SET
> > > > hence you missed the point completely.

> > > Lists ARE countable sets, though not all sets need to be lists.

> > This PROVES you MISSED THE POINT!

> > Countable SETS are NOT LISTS.

> But their members necessarily are capable of being listed by the very
> proof that they are countable.

> > This set has no diagonal.

> > 0.102  0.434  0.543
> > 0.245  0.545  0.544
> > 0.095  0.111  0.999

> > Herc

> 0.102000000
> 0.434000000
> 0.543000000
> 0.245000000
> 0.545000000
> 0.544000000
> 0.095000000
> 0.111000000
> 0.999000000

> Now it does!

> > In article
> > <acc4f4cb-7f4d-4a0d-bb27-c9682b41e...@r10g2000pbd.googlegroups.com>,
> >   Graham Cooper<grahamcoop...@gmail.com>  wrote:

> >> On Nov 12, 10:23 am, forbisga...@gmail.com wrote:
> >>> On Sunday, November 11, 2012 1:59:42 PM UTC-8, Graham Cooper wrote:
> >>>> like looking at the position of an electron with a microscope and
> >>>> claiming the position is fixed.

> >>> The problem is Cantor dealt with a well ordered set.  A well ordered
> >>> set has a fixed index.  The same number will alway appear at the same
> >>> index.  You appear to be claiming there is no fixed index for a well
> >>> ordered set.

> >> I claim 20 flaws in Cantor's proof!

> > A lot of nuts have claimed flaws in Cantor's proofs, but, as yet, none
> > of those nuts has proved not to be cracked.

> [...]

> I think one ontological (non-mathematical)
> problematic is that if something can't be
> listed, enumerated, then it can't be shown (like
> in "show the story", don't just "tell the story"
> in journalism).

> The story of the reals:  if it could be shown completely
> in a movie, that would be enumerable.

> The non-believers are not satisfied with a logical proof ...

> Dave

> a) N <-BIJECTS-> GODEL NUMBERS

> b) GODEL NUMBERS <-BIJECT-> FUNCTIONS

> c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS

> c) CHOICE FUNCTIONS <-BIJECT-> SETS

> d) |SETS| > |N|
> *****************************************************
> *****************************************************

> > Q5
> > Which 1 of these does not hold?

> > a) N <-BIJECTS-> GODEL NUMBERS

> > b) GODEL NUMBERS <-BIJECT-> FUNCTIONS

> > c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS

> > c) CHOICE FUNCTIONS <-BIJECT-> SETS

> > d) |SETS| > |N|
> > *****************************************************
> > *****************************************************

> b) is false. What is true is that

>  GODEL NUMBERS <-BIJECT-> CONSTRUCTABLE FUNCTIONS

> However

>       CONSTRUCTABLE FUNCTIONS <-NOT BIJECT-> CHOICE FUNCTIONS

> > On Nov 12, 5:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >  *****************************************************

> > > Q5
> > > Which 1 of these does not hold?

> > > a) N <-BIJECTS-> GODEL NUMBERS

> > > b) GODEL NUMBERS <-BIJECT-> FUNCTIONS

> > > c) FUNCTIONS <-BIJECT-> CHOICE FUNCTIONS

> > > c) CHOICE FUNCTIONS <-BIJECT-> SETS

> > > d) |SETS| > |N|
> > > *****************************************************
> > > *****************************************************

> > b) is false. What is true is that

> >  GODEL NUMBERS <-BIJECT-> CONSTRUCTABLE FUNCTIONS

> > However

> >       CONSTRUCTABLE FUNCTIONS <-NOT BIJECT-> CHOICE FUNCTIONS

> So what's an unlistable choice function example?