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From: |-|ercules on 9 Jun 2010 01:52
(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?

Herc
--
the nonexistence of a box that contains the numbers of all the boxes
that don't contain their own box number implies higher infinities.
- Cantor's Proof (the holy grail of paradise in mathematics)
From: Daryl McCullough on 9 Jun 2010 06:57
|-|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?

--
Daryl McCullough
Ithaca, NY

From: |-|ercules on 9 Jun 2010 07:17
"Daryl McCullough" <stevendaryl3016(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!

Here is what is ACTUALLY happening.

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

Herc

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
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