How Wide Do All The Possible Computed Digit Sequences Go? [Math]

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From: Sylvia Else on 22 Jun 2010 22:23

On 23/06/2010 12:20 PM, Graham Cooper wrote:
> On Jun 23, 10:48 am, Sylvia Else<syl...(a)not.here.invalid> wrote:
>> On 23/06/2010 6:17 AM, Graham Cooper wrote:
>>
>>
>>
>>
>>
>>> On Jun 22, 9:56 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>> On 22/06/2010 8:13 PM, Graham Cooper wrote:
>>
>>>>> On Jun 22, 8:04 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>> On 22/06/2010 7:39 PM, Graham Cooper wrote:
>>
>>>>>>> On Jun 22, 7:33 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>>>> On 22/06/2010 7:21 PM, Graham Cooper wrote:
>>
>>>>>>>>> On Jun 22, 7:14 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>>>>>> On 22/06/2010 6:30 PM, Graham Cooper wrote:
>>
>>>>>>>>>>> On Jun 22, 6:19 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>>>>>>>> On 22/06/2010 6:14 PM, Graham Cooper wrote:
>>
>>>>>>>>>>>>> On Jun 22, 6:05 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>>>>>>>>>> On 22/06/2010 5:52 PM, Graham Cooper wrote:
>>
>>>>>>>>>>>>>>> On Jun 22, 5:48 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>>>>>>>>>>>>>>>> On 22/06/2010 5:06 PM, Graham Cooper wrote:
>>
>>>>>>>>>>>>>>>>> On Jun 22, 4:33 pm, Rupert<rupertmccal...(a)yahoo.com> wrote:
>>>>>>>>>>>>>>>>>> There does not exist an ordinal number x, such that the set of all
>>>>>>>>>>>>>>>>>> sequences of decimal digits of length x has cardinality aleph-null.
>>>>>>>>>>>>>>>>>> However, the set of all *computable* sequences of decimal digits of
>>>>>>>>>>>>>>>>>> length aleph-null does have cardinality aleph-null. But it is not
>>>>>>>>>>>>>>>>>> equal to the set of *all* sequences of decimal digits of length aleph-
>>>>>>>>>>>>>>>>>> null.
>>
>>>>>>>>>>>>>>>>> So you are disputing the formula 10^x reals can list
>>>>>>>>>>>>>>>>> all digit permutations x digits wide?
>>
>>>>>>>>>>>>>>>> He didn't say that at all. How on Earth did you get there?
>>
>>>>>>>>>>>>>>>> Sylvia.
>>
>>>>>>>>>>>>>>> The question I gave him was an application of that formula
>>>>>>>>>>>>>>> his answer was not.
>>
>>>>>>>>>>>>>> I dare say, but your suggested inference was still not valid. His answer
>>>>>>>>>>>>>> said nothing about what 10^x reals can do.
>>
>>>>>>>>>>>>>> Sylvia.
>>
>>>>>>>>>>>>> What kind of muddled logic is that?
>>
>>>>>>>>>>>> Well, did his answer say something about what 10^x reals can do? If so,
>>>>>>>>>>>> what did it say? Where did it say it?
>>
>>>>>>>>>>>> Sylvia.
>>
>>>>>>>>>>> Huh? He didn't use the the formula to answer the question
>>>>>>>>>>> so I said he must be disputing the formula. As the answer is
>>>>>>>>>>> a simple application of the formula.
>>
>>>>>>>>>> It's hardly a simple application. For a start, your question was phrased
>>>>>>>>>> the other way around, so that a logarithm to base 10 and ceiling
>>>>>>>>>> function would be required for a finite set of numbers. But you can't
>>>>>>>>>> just plug infinity into functions that are valid for finite arguments,
>>>>>>>>>> and expect to get a meaningful answer, and it's not surprising that
>>>>>>>>>> Rupert didn't try.
>>
>>>>>>>>>>> If you're going to disagree with me say opposing statements
>>>>>>>>>>> this is very confusing where you're going, as predicted
>>
>>>>>>>>>> What does that mean? Why does your ability to express yourself in
>>>>>>>>>> English take these turns for the worse?
>>
>>>>>>>>>> Sylvia.
>>
>>>>>>>>> So if y = log (x)
>>>>>>>>> and x = infinity
>>
>>>>>>>> False proposition.
>>
>>>>>>>>> you don't know y ?
>>
>>>>>>>> Nothing to know - see above.
>>
>>>>>>>>> You have 1000 theorems of transfiniteness but can't
>>>>>>>>> do sums with infinity?
>>
>>>>>>>> Sums are not defined with infinity.
>>
>>>>>>>> Sylvia.
>>
>>>>>>> You are reaching.
>>
>>>>>>> What is false?
>>
>>>>>>> Y = log (x)
>>
>>>>>>> or
>>
>>>>>> y = log (x) and x = infinity.
>>
>>>>>> That statement is false.
>>
>>>>>> Sylvia.
>>
>>>>> Why? Ignoring your other copious bullshit.
>>
>>>> The function log(x) is not defined for x = infinity, so whatever value y
>>>> has, it cannot possibly equal the result from the log function.
>>
>>>> The nearest you can get is that y tends to infinity as x tends to infinity.
>>
>>>> Sylvia.
>>
>>> So as number of reals in the list of computable reals tends to oo
>>> the digit width of 'every permutation' tends to infinity.
>>
>>> BUT if the number of computable reals WAS oo the digit width
>>> of 'every permutation' is NOT infinity.
>>
>> You're applying the reasoning backwards.
>>
>> An incorrect line of reasoning can lead to a correct result.
>>
>> For example, I allege the following to be a theorem:
>>
>> 2 + 2 = 4, therefore sqrt(9) = 3.
>>
>> You can dismantle the reasoning, and show that it is not a theorem. but
>> sqrt(9) is still 3.
>>
>>
>>
>>> What abou this
>>
>>> x = oo
>>> y = x
>>
>>> what is y?
>>
>> I have no problem with y being oo (provided the = operator is transitive
>> in the particular set of axioms involved).
>>
>> Sylvia.
>
>
> What about
>
> y = 1 * x
> x = oo
>
> what is y?
>
> Herc
>

Depends what your axioms have to say about the special case of
multiplying by 1.

Sylvia.

From: Sylvia Else on 22 Jun 2010 22:29

On 23/06/2010 12:20 PM, Graham Cooper wrote:

BTW, what are you going to do about my demolition of herc_cant_3?

Sylvia.

From: Graham Cooper on 22 Jun 2010 22:35

On Jun 23, 12:29 pm, Sylvia Else <syl...(a)not.here.invalid> wrote:
> On 23/06/2010 12:20 PM, Graham Cooper wrote:
>
> BTW, what are you going to do about my demolition of herc_cant_3?
>
> Sylvia.

Huh? I just LOLed does that count?

You think there's bigger sets than oo
based on the fact

Log(oo) =\= oo

atleast you understand that all permutations
of infinite width in a countable list would
disprove transfiniteness as you are coming up
with numerous stories to deny the fact.

Herc

From: Sylvia Else on 22 Jun 2010 22:51

On 23/06/2010 12:35 PM, Graham Cooper wrote:
> On Jun 23, 12:29 pm, Sylvia Else<syl...(a)not.here.invalid> wrote:
>> On 23/06/2010 12:20 PM, Graham Cooper wrote:
>>
>> BTW, what are you going to do about my demolition of herc_cant_3?
>>
>> Sylvia.
>
>
>
> Huh? I just LOLed does that count?

No. You've tried it before, and it didn't work then either.

To recap, herc_cant_3 was based on the proposition that if a list
contains all finite prefixes, then it contains all infinite sequences,
and thus all reals.

However all finite prefixes can be obtained by taking a list of all
computables, permuting it and taking the diagonals. The set of such
diagonals thus contains all finite prefixes. However, it cannot contain
all infinite sequences because the list of all computables contains
0.111.... meaning that every member of the set must contain a 1.

Thus it is not true that if a list contains all finite prefixes, it
contains all infinite sequences and thus all reals.

Herc-cant_3 is therefore not a theorem.

Sylvia.

From: Tim Little on 22 Jun 2010 23:01

On 2010-06-23, Sylvia Else <sylvia(a)not.here.invalid> wrote:
> To recap, herc_cant_3 was based on the proposition that if a list
> contains all finite prefixes, then it contains all infinite
> sequences, and thus all reals.
>
> However all finite prefixes can be obtained by taking a list of all
> computables, permuting it and taking the diagonals. [...]

An even more straightforward counterexample was given much earlier in
the discussion: a list of all finite digit sequences obviously
contains all finite prefixes, and by definition does not contain *any*
infinite sequences. So it certainly does not contain all of them.

Herc didn't accept that one, so I doubt he'll accept (or even
understand) yours.

- Tim