 | | From: | |-|erc | | Subject: | "A random real number will be on a computables list to an infinite number of digits" | | Date: | Thu, 20 Jan 2005 11:43:01 +1000 |
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 | True / False / Other ?
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"All finite subsequences of a random real number will be on a computables list" True / False / Other
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"All digits of a random real number are covered in all finite subsequences of that number" True / False / Other
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"If you have the list of computables, a random real number can be on it to an infinite number
of digits, and yet not be on the list" True / False / Other
___
Herc
--
Have you now or have you ever been a member of the antidisestablishmentarianism party?
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| | From: | Will Twentyman | | Subject: | Re: "A random real number will be on a computables list to an infinite | | Date: | Thu, 20 Jan 2005 12:31:57 -0500 |
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| Note: where you use the word "random", I will interpret it as
"arbitrary". I suspect the answers will be satisfactory.
|-|erc wrote:
> True / False / Other ?
> ____
>
>
> "All finite subsequences of a random real number will be on a computables list" True / False / Other
Other: it depends on which computables list.
> ____
>
>
> "All digits of a random real number are covered in all finite subsequences of that number" True / False / Other
Other: what do you mean by "covered"?
> ____
>
>
> "If you have the list of computables, a random real number can be on it to an infinite number
> of digits, and yet not be on the list" True / False / Other
False: a number is either on the list or not. However, if you had
asked: "If you have a list of computables, an arbitrary real number can
have an arbitrarily long prefix appear on the list, yet the real number
not be on the list." the answer would be True.
--
Will Twentyman
email: wtwentyman at copper dot net
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| | From: | George Cox | | Subject: | Re: "A random real number will be on a computables list to an infinite | | Date: | Thu, 20 Jan 2005 02:50:56 +0000 (UTC) |
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| |-|erc wrote:
>
> True / False / Other ?
> ____
>
> "All finite subsequences of a random real number
What is a random real number?
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| | From: | The Ghost In The Machine | | Subject: | Re: "A random real number will be on a computables list to an infinite number of digits" | | Date: | Thu, 20 Jan 2005 15:00:18 GMT |
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| In sci.logic, George Cox
wrote
on Thu, 20 Jan 2005 02:50:56 +0000 (UTC)
<41EF1C98.718BD804@spambtinternet.com.invalid>:
> |-|erc wrote:
>>
>> True / False / Other ?
>> ____
>>
>> "All finite subsequences of a random real number
>
> What is a random real number?
The quesion
"All finite subsequences of a random real number
are in a computable list L"
can be rerendered:
"What is the probability that, for an arbitrary element r of R,
that Prefix(r) is wholly contained in the computable list L?"
where Prefix(r) = {q: q = floor[r * 10^n]/10^n, n >= 0, n in J}.
AFAIK, this is a fairly general technique for dealing with
probability questions -- an extension of T/F questions.
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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| | From: | |-|erc | | Subject: | Re: "A random real number will be on a computables list to an infinite number of digits" | | Date: | Thu, 20 Jan 2005 13:07:55 +1000 |
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| "George Cox" wrote in
> |-|erc wrote:
> >
> > True / False / Other ?
> > ____
> >
> > "All finite subsequences of a random real number
>
> What is a random real number?
there is no known pattern to the digits.
Herc
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| | From: | George Cox | | Subject: | Re: "A random real number will be on a computables list to an infinite | | Date: | Thu, 20 Jan 2005 19:01:58 +0000 (UTC) |
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| |-|erc wrote:
>
> "George Cox" wrote in
> > |-|erc wrote:
> > >
> > > True / False / Other ?
> > > ____
> > >
> > > "All finite subsequences of a random real number
> >
> > What is a random real number?
>
> there is no known pattern to the digits.
There is no known pattern[1] in pi. Is pi random?
[1] Depends on what one means by pattern of course.
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| | From: | |-|erc | | Subject: | Re: "A random real number will be on a computables list to an infinite number of digits" | | Date: | Fri, 21 Jan 2005 12:53:22 +1000 |
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| "George Cox" wrote
> |-|erc wrote:
> >
> > "George Cox" wrote in
> > > |-|erc wrote:
> > > >
> > > > True / False / Other ?
> > > > ____
> > > >
> > > > "All finite subsequences of a random real number
> > >
> > > What is a random real number?
> >
> > there is no known pattern to the digits.
>
> There is no known pattern[1] in pi. Is pi random?
>
> [1] Depends on what one means by pattern of course.
pi's sequence of digits is well known
Herc
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| | From: | The Ghost In The Machine | | Subject: | Re: "A random real number will be on a computables list to an infinite number of digits" | | Date: | Fri, 21 Jan 2005 05:01:42 GMT |
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| In sci.logic, |-|erc
wrote
on Fri, 21 Jan 2005 12:53:22 +1000
<35b94gF4gjg8vU1@individual.net>:
> "George Cox" wrote
>> |-|erc wrote:
>> >
>> > "George Cox" wrote in
>> > > |-|erc wrote:
>> > > >
>> > > > True / False / Other ?
>> > > > ____
>> > > >
>> > > > "All finite subsequences of a random real number
>> > >
>> > > What is a random real number?
>> >
>> > there is no known pattern to the digits.
>>
>> There is no known pattern[1] in pi. Is pi random?
>>
>> [1] Depends on what one means by pattern of course.
>
> pi's sequence of digits is well known
>
And even computable. Is it periodic? No.
Is it normal? No one's proven it either way AFAIK,
though I for one suspect it is.
Is it rational? No.
Personally, I prefer changing questions such as:
"All finite subsequences of a random real number are in the list L"
to something like
"The measure of the intersection of the set of all finite subsequences
of real numbers intersect the list L".
or perhaps
"The measure of the intersection of the set of all finite subsequences
of real numbers intersect the set of all finite subsequences of
elements in L".
(The problem with the latter is that it will only result in subsets
of TX_10 = {k/10^n: n >= 0, k,n in J}. This set does not cover R,
or even Q.)
In most cases, a "random real number" question is really
an "arbitrary real number" question.
> Herc
>
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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| | From: | The Ghost In The Machine | | Subject: | Re: "A random real number will be on a computables list to an infinite number of digits" | | Date: | Thu, 20 Jan 2005 15:00:17 GMT |
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| Subject added to post for clarity.
Assumptions:
[1] Lebesgue measure theory standard metric for R.
[2] Standard base-10.
[3] The number of machines able to compute a number is
countably infinite as the mapping N -> M(N)
is 1-1 but not necessarily onto.
In sci.logic, |-|erc
wrote
on Thu, 20 Jan 2005 11:43:01 +1000
<358gk5F4jesvdU1@individual.net>:
> "A random real number will be on a computables list
> to an infinite number of digits"
> True / False / Other ?
Probability 0.00%.
[Measure of select set: 0]
[Measure of other set: infinite]
> ____
>
>
> "All finite subsequences of a random real number will be
> on a computables list" True / False / Other
Probability 0.00%.
[Measure of select set: 0]
[Measure of other set: infinite]
> ____
>
>
> "All digits of a random real number are covered in
> all finite subsequences of that number" True / False / Other
Exception: illegal instruction
Op: comparison
Operand1: digit
Operand2: real
Operation aborted.
> ____
>
>
> "If you have the list of computables, a random real number
> can be on it to an infinite number of digits, and yet not
> be on the list" True / False / Other
The question is assumed to be:
"If one has a computable set L over R and a random real number r,
what is the probability of r in L,
given Prefix(r) intersection L being Prefix(r)?"
Probability 0.00%.
[Measure of select set: 0]
[Measure of other set: infinite]
[.sigsnip]
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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