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From: William Hughes on 9 Jun 2010 07:53
On Jun 9, 8:17 am, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> "Daryl McCullough" <stevendaryl3...(a)yahoo.com> wrote...
> > |-|ercules says...
>
> >>(from the "Xenides dies" thread)
>
> >>> For *all* N, the sequence differs from the Nth entry in the list at
> >>> the Nth digit (and possibly other positions as well).  It is new
> >>> because for *every* sequence in the list, the question "is it the same
> >>> as this sequence" is answered "no".
>
> >>So you think the antidiagonal comes up with an actual NEW SEQUENCE OF DIGITS
> >>and this does not contradict that ALL sequences of digits are on the computable
> >>list of reals up to all (an infinite amount of) digit positions?
>
> > You start with a completely crystal clear statement:
> > The antidiagonal number is not equal to any number on the list.
>
> > Then you paraphrase this clear statement to get a completely
> > muddled statement:
>
> >>the antidiagonal comes up with an actual NEW SEQUENCE OF DIGITS
> >>and this does not contradict that ALL sequences of digits are on
> >>the computable list of reals up to all (an infinite amount of)
> >>digit positions
>
> > Why do you prefer to use muddled, incoherent statements instead of
> > clear ones?
>
> > The antidiagonal is not equal to any of the numbers on the list.
> > What is unclear about that?
>
> It's based on this argument.
>
> 123
> 456
> 789
>
> DIAG = 159
> ANTIDIAG = 260
>
> 260 is not on the list, it's a NEW DIGIT SEQUENCE.
>
> You claim this works on infinite lists.
>
> You claim no list contains EVERY DIGIT SEQUENCE
> because you can find a NEW DIGIT SEQUENCE
>
> But the computable real list contains EVERY DIGIT SEQUENCE
> up to all (an infinite amount of) finite lengths.
>
> EVERY DIGIT SEQUENCE POSSIBLE UP TO INFINITY!
>
> BELOW IS A *VALID* DIAGONAL ARGUMENT
>
> 123
> 456
> 789
>
> DIAG = 159
> ANTIDIAG = 260
>
> See how it actually generates a NEW SEQUENCE OF DIGITS!!!!!!!!!!!!!!!!
>
> Your argument doesn't do that!

Yes it does. If you start with a list that exists
you get a new sequence of digits. (E.g. if you
start with a list of sequences, every one of
which has a last digit, you get a sequence that
does not have a last digit).

>
> Here is what is ACTUALLY happening.
>

0. Assume a list containing all sequences exists

> 1 Start with a list containing all sequences.
> 2 Find a NEW sequence
> 3 CONTRADICTION

4. Conclude that the assumption is false.

- William Hughes
From: Daryl McCullough on 9 Jun 2010 09:29
|-|ercules says...
>
>"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote...

>> The antidiagonal is not equal to any of the numbers on the list.
>> What is unclear about that?

>It's based on this argument.
>
>123
>456
>789
>
>DIAG = 159
>ANTIDIAG = 260
>
>260 is not on the list, it's a NEW DIGIT SEQUENCE.
>
>You claim this works on infinite lists.

Yes, it does. The antidiagonal is not equal to the
first number on the list, because its first digit
is different from the first digit of the first number.
It is different from the second number, because it has
a different second digit. It is different from the
third number, because it has a different third digit.

We can prove, in general:

Lemma:

Given any list of reals L,
there exists a real r,
such that r is not on the list L.

What is it that you don't understand?

An immediate consequence of this lemma is:

Theorem: There is no list that contains every real number.

>Here is what is ACTUALLY happening.
>
>1 Start with a list containing all sequences.
>2 Find a NEW sequence
>3 CONTRADICTION

You assume that there exists a list containing
all sequences, and then you find out that assumption
leads to a contradiction. So the assumption is false.

--
Daryl McCullough
Ithaca, NY

From: |-|ercules on 9 Jun 2010 11:01
"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote...
> |-|ercules says...
>>
>>"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote...
>
>>> The antidiagonal is not equal to any of the numbers on the list.
>>> What is unclear about that?
>
>>It's based on this argument.
>>
>>123
>>456
>>789
>>
>>DIAG = 159
>>ANTIDIAG = 260
>>
>>260 is not on the list, it's a NEW DIGIT SEQUENCE.
>>
>>You claim this works on infinite lists.
>
> Yes, it does. The antidiagonal is not equal to the
> first number on the list, because its first digit
> is different from the first digit of the first number.
> It is different from the second number, because it has
> a different second digit. It is different from the
> third number, because it has a different third digit.
>
> We can prove, in general:
>
> Lemma:
>
> Given any list of reals L,
> there exists a real r,
> such that r is not on the list L.
>
> What is it that you don't understand?
>
> An immediate consequence of this lemma is:
>
> Theorem: There is no list that contains every real number.
>
>>Here is what is ACTUALLY happening.
>>
>>1 Start with a list containing all sequences.
>>2 Find a NEW sequence
>>3 CONTRADICTION
>
> You assume that there exists a list containing
> all sequences, and then you find out that assumption
> leads to a contradiction. So the assumption is false.
>

not necessarily. I demonstrated that all sequences occur and your 'new sequence'
is merely self reference and negation.

Are you saying you find a new_sequence_of_digits using diagonalization on infinite lists?

Like 260?

How so when ALL (INFINITELY MANY) digits of ALL digit sequences are computable?

Herc
From: WM on 9 Jun 2010 11:08
On 9 Jun., 15:29, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:
> |-|ercules says...
>
>
>
>
>
>
>
> >"Daryl McCullough" <stevendaryl3...(a)yahoo.com> wrote...
> >> The antidiagonal is not equal to any of the numbers on the list.
> >> What is unclear about that?
> >It's based on this argument.
>
> >123
> >456
> >789
>
> >DIAG = 159
> >ANTIDIAG = 260
>
> >260 is not on the list, it's a NEW DIGIT SEQUENCE.
>
> >You claim this works on infinite lists.
>
> Yes, it does. The antidiagonal is not equal to the
> first number on the list, because its first digit
> is different from the first digit of the first number.
> It is different from the second number, because it has
> a different second digit. It is different from the
> third number, because it has a different third digit.
>
> We can prove, in general:
>
> Lemma:
>
> Given any list of reals L,
> there exists a real r,
> such that r is not on the list L.
>
> What is it that you don't understand?
>
> An immediate consequence of this lemma is:
>
> Theorem: There is no list that contains every real number.

There is a list that contains every real number that can be contained
in a list:

0
1
00
01
10
11
000
....

Some people claim that further real numbers could be defined by
infinite words (sequences). But that is wrong.
It is true that a finite definition like "0,111..." or "pi" defines
every digit of an infinite sequence and hence a real number.
Finite definition ==> Infinite sequence
But it is wrong to switch this implcation to
Infinite sequence ==> Finite definition

The real number would only be defined if the last term of the infinite
sequence was defined. But that is impossible because there is no last
term.

Regards, WM
From: Daryl McCullough on 9 Jun 2010 12:54
|-|ercules says...
>
>"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote...

>>>1 Start with a list containing all sequences.
>>>2 Find a NEW sequence
>>>3 CONTRADICTION
>>
>> You assume that there exists a list containing
>> all sequences, and then you find out that assumption
>> leads to a contradiction. So the assumption is false.
>
>not necessarily. I demonstrated that all sequences occur and your
>'new sequence' is merely self reference and negation.

There is no self-reference in the definition of the anti-diagonal
function. The anti-diagonal function is a two-place function

f(L,n)

which returns a digit for each list L and for each natural number
n. It's a simple function:

First define a transformation on digits c(d) as follows:
c(5) = 4. If d is not equal to 5, then c(d) = 5.

Now, we define f(L,n) as follows:

f(L,n) = c(L(n,n))

where L(n,n) = the nth digit of the nth real in the list L.

Now, the antidiagonal real is defined by:

antiDiag(L) = that real r such that the integer part of r is 0,
and forall n, the nth digit of r is equal to f(L,n).

This is a function that given any list of reals L, returns
another real, antiDiag(L), which is guaranteed to not be on
the list L.

There is nothing self-referential about this definition.

>Are you saying you find a new_sequence_of_digits using
>diagonalization on infinite lists?

Yes, I am absolutely saying that. Let's try it out with
a simple example.

Let's let L be the list containing the falling reals:

L_0 = 0.000...
L_1 = 0.300...
L_2 = 0.3300...
L_3 = 0.3330...
etc.

Then the antidiagonal will be
0.555...

This is clearly not on list L.

You can try yourself with different choices for the list L, as
long as it is possible to determine what the first real is,
what the second real is, etc.

--
Daryl McCullough
Ithaca, NY

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