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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

From: Daryl McCullough on 9 Jun 2010 13:03
|-|ercules says...

>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?

Yes, all finite sequences of digits are computable. So let's
go ahead and assume that we have a list that includes every
finite sequence of digits. For definiteness, let's use the following
rule: Let real number n be the decimal representation of n, preceded
by a decimal point. So, we have:

L_0 = 0.0
L_1 = 0.1
L_2 = 0.2
....
L_10 = 0.10
L_11 = 0.11
....
L_100 = 0.100
....

The way I've enumerated them, some numbers appear more than once. For
example, L_1 is equal (as a real number) to L_10. But that doesn't matter.

With this list, every finite decimal expansion occurs somewhere on the list.
But not every *infinite* decimal expansion occurs. So let's apply our
diagonalization procedure: we get

r = 0.5555555555...

This number is not on the list. It's not equal to L_0, it's not equal
to L_1, it's not equal to L_2, etc.

The finite approximations are on the list, however. 0.5 is equal to
L_5. 0.55 is equal to L_55. 0.555 is equal to L_555. etc. But the
full number 0.555... (which happens to be the decimal representation
of the fraction 5/9) is not anywhere on the list.

--
Daryl McCullough
Ithaca, NY

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