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 | | From: | |-|erc | | Subject: | HERC 97 SCI.MATH 0 | | Date: | Thu, 20 Jan 2005 17:19:22 +1000 |
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 | 3 more propositions, see if you can work out the truth value.
THE PROBLEM
(countable) infinite people each flip coins (countable) infinite times each.
Can you always come up with a new coin sequence?
SCI.MATH SOLUTION
Take the inverted diagonal of the flippers. Call this diagonal.
PROPOSITION 1
"Regarding the infinite set of people flipping coins,
the diagonal coin sequence has been flipped only to some finite amount of flips of the diagonal"
PROPOSITION 2
"Regarding the infinite set of people flipping coins,
the diagonal coin sequence has been flipped to an infinite amount of flips of the diagonal"
PROPOSITION 3
"Regarding the infinite set of people flipping coins,
the diagonal coin sequence has been flipped some finite amount (or 0) of flips per person"
Herc
--
Stardate 20 1 2005. Its becoming obvious the reason every thread results in argument is because half
of people are outright lying. Putting contradictory remarks side by side takes an average of 10 posts before
they accept it's known they were caught.
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| | From: | The Ghost In The Machine | | Subject: | Re: HERC 97 SCI.MATH 0 | | Date: | Thu, 20 Jan 2005 15:00:20 GMT |
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| In sci.logic, |-|erc
wrote
on Thu, 20 Jan 2005 17:19:22 +1000
<3594arF4hqdmkU1@individual.net>:
> 3 more propositions, see if you can work out the truth value.
>
> THE PROBLEM
> (countable) infinite people each flip coins (countable) infinite
> times each. Can you always come up with a new coin sequence?
>
> SCI.MATH SOLUTION
> Take the inverted diagonal of the flippers. Call this diagonal.
>
>
> PROPOSITION 1
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped only to some finite
> amount of flips of the diagonal"
>
> PROPOSITION 2
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped to an infinite amount
> of flips of the diagonal"
>
> PROPOSITION 3
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped some finite amount
> (or 0) of flips per person"
None of these make sense as written. Did you mean something
along the lines of:
"Given the diagonal flip sequence, indicate the following:
[1] The diagonal flip sequence is somewhere on the list of
infinite sequences.
[2] All prefixes of the diagonal flip sequence are embedded somewhere
on the prefixes of the list of infinite sequences.
[3] Up to a count N, all prefixes of length at most N are embedded
somewhere on the prefixes of the list of infinite sequences."
?
Also, we'll need more data on the people's flips. For all I know
every one of them is given a coin that does nothing but tails. :-)
[.sigsnip]
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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| | From: | ken quirici | | Subject: | Re: HERC 97 SCI.MATH 0 | | Date: | 20 Jan 2005 07:49:23 -0800 |
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| |-|erc wrote:
> 3 more propositions, see if you can work out the truth value.
>
> THE PROBLEM
> (countable) infinite people each flip coins (countable) infinite
times each.
> Can you always come up with a new coin sequence?
>
I think there's something interesting here, which I will get to
presently, but first, to answer the above question:
Yes, clearly. Any countable list of countably many H's and T's has a
constructible 'diagonal' which is different from any member of the
list. So what? You haven't specified anything 'interesting' about this
list of infinite sequences of flips. For example, the list could be all
the same flipping sequence. If you mean the lists to be random, then
again, the lists have nothing 'interesting' about them. Sure, for some
random countable list of flip sequences I can generate the diagonal.
Again, so what?
> SCI.MATH SOLUTION
> Take the inverted diagonal of the flippers. Call this diagonal.
>
>
> PROPOSITION 1
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped only to some finite
amount of flips of the diagonal"
>
> PROPOSITION 2
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped to an infinite amount of
flips of the diagonal"
>
> PROPOSITION 3
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped some finite amount (or 0)
of flips per person"
>
> Herc
Let's assume we have a more interesting list of sequences - the list
of all possible flipping sequences. Then the diagonalization argument
shows that that list is not countable, by finding a list not in the
sequence. Given that there's a bijection between your flipping lists
and binary numbers and therefore with the integers, this is not
surprising. Interesting but not the Holy Grail, and not really
answering
your questions.
But now we come to your more interesting question of how much of any
diagonal constructed from some list purporting to be all possible
flipping sequences, match the same number of initial digits of some
one of the flipping lists. For example, is there some flipping
sequence whose first 17 digits match the first 17 digits of the
diagonal, say maybe sequence #1789723? And, what is the LARGEST number
of such digits for which we can find a matching element of the list of
flipping sequences?
I think that's a fair paraphrase of your 3-choice problem.
As far as an answer, I haven't the foggiest but I think it depends on
how you order your list of flipping sequences. If you choose a kind
of lexicographical order, where after 2, you have all possible 1-flip
sequences, and after another 4, you have all possible 2-flip sequences,
etc., then the answer would be, for any n however large, I can find
one of the flipping sequences that match the first n digits of the
diagonal.
My CLAIM is that Herc's questions, given an 'interesting' list of
flipping sequences, depend on the ordering of the list, and hence
go beyond the simpler question of denumerability. But I'm uneasy
with this conclusion, probably for good reason.
Thanks.
Ken
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| | From: | Will Twentyman | | Subject: | Re: HERC 97 SCI.MATH 0 | | Date: | Thu, 20 Jan 2005 15:00:03 -0500 |
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| |-|erc wrote:
> 3 more propositions, see if you can work out the truth value.
>
> THE PROBLEM
> (countable) infinite people each flip coins (countable) infinite times each.
> Can you always come up with a new coin sequence?
Yes.
> SCI.MATH SOLUTION
> Take the inverted diagonal of the flippers. Call this diagonal.
>
>
> PROPOSITION 1
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped only to some finite amount of flips of the diagonal"
True, if you include 0 as a possibility. For example, all the people
could flip heads, then the first result on the diagonal is tails and its
longest prefix that was flipped is 0 flips long.
>
> PROPOSITION 2
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped to an infinite amount of flips of the diagonal"
False, though it is possible that every finite prefix of diagonal is one
of the prefixes of one of the flippers.
> PROPOSITION 3
> "Regarding the infinite set of people flipping coins,
> the diagonal coin sequence has been flipped some finite amount (or 0) of flips per person"
True.
--
Will Twentyman
email: wtwentyman at copper dot net
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| | From: | |-|erc | | Subject: | Re: HERC 97 SCI.MATH 0 | | Date: | Fri, 21 Jan 2005 12:19:32 +1000 |
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| "Will Twentyman" wrote in
> |-|erc wrote:
>
> > 3 more propositions, see if you can work out the truth value.
> >
> > THE PROBLEM
> > (countable) infinite people each flip coins (countable) infinite times each.
> > Can you always come up with a new coin sequence?
>
> Yes.
>
> > SCI.MATH SOLUTION
> > Take the inverted diagonal of the flippers. Call this diagonal.
> >
> >
> > PROPOSITION 1
> > "Regarding the infinite set of people flipping coins,
> > the diagonal coin sequence has been flipped only to some finite amount of flips of the diagonal"
>
> True, if you include 0 as a possibility. For example, all the people
> could flip heads, then the first result on the diagonal is tails and its
> longest prefix that was flipped is 0 flips long.
Is this always the case? we are regarding all possible cases "can you ALWAYS come up with a new sequence"
for example, the coin flips could be all the halting rows of UTM(person, flip) mod 2.
>
> >
> > PROPOSITION 2
> > "Regarding the infinite set of people flipping coins,
> > the diagonal coin sequence has been flipped to an infinite amount of flips of the diagonal"
>
> False, though it is possible that every finite prefix of diagonal is one
> of the prefixes of one of the flippers.
You must ALWAYS be able to form a new sequence, (i.e. for ANY possible list of flippers),
you have to consider that possibility.
If all prefixes of the diagonal match prefixes of the flippers,
then what portion of the diagonal is contained in all its prefixes? finite or infinite?
What portion of the diagonal is matched on the list of flippers? finite or infinite?
>
> > PROPOSITION 3
> > "Regarding the infinite set of people flipping coins,
> > the diagonal coin sequence has been flipped some finite amount (or 0) of flips per person"
>
> True.
>
> --
> Will Twentyman
> email: wtwentyman at copper dot net
The question is simple. You ASSUME a good random listing is achieved.
Assume only a finite number of the diags flips occur in initial segments of the flippers outcomes.
Then this segment has length L.
There are 2^L possible initial segments of that length.
Therefore, with oo flippers being infinitely larger than 2^L, it is improbable for each one
of the 2^L initial segments not to have been flipped.
Therefore, there is no such L.
This case is ATLEAST one of the possibilites you have to consider to ALWAYS find a new sequence.
THEN, THE QUESTION IS ABOUT THE NEW SEQUENCE.
What portion of the new sequence are part of initial segments of the flippers?
A FINITE PORTION
INFINITE
Herc
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| | From: | Will Twentyman | | Subject: | Re: HERC 97 SCI.MATH 0 | | Date: | Thu, 20 Jan 2005 22:13:21 -0500 |
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| |-|erc wrote:
> "Will Twentyman" wrote in
>
>>|-|erc wrote:
>>
>>
>>>3 more propositions, see if you can work out the truth value.
>>>
>>>THE PROBLEM
>>>(countable) infinite people each flip coins (countable) infinite times each.
>>>Can you always come up with a new coin sequence?
>>
>>Yes.
>>
>>
>>>SCI.MATH SOLUTION
>>>Take the inverted diagonal of the flippers. Call this diagonal.
>>>
>>>
>>>PROPOSITION 1
>>>"Regarding the infinite set of people flipping coins,
>>>the diagonal coin sequence has been flipped only to some finite amount of flips of the diagonal"
>>
>>True, if you include 0 as a possibility. For example, all the people
>>could flip heads, then the first result on the diagonal is tails and its
>>longest prefix that was flipped is 0 flips long.
>
> Is this always the case? we are regarding all possible cases "can you ALWAYS come up with a new sequence"
> for example, the coin flips could be all the halting rows of UTM(person, flip) mod 2.
1) "For each flipper, the prefix in common with diagonal is finitely
long." Under that interpretation: yes. This is the interpretation I
had in mind when answering.
2) "For the diagonal, there is a maximum length prefix for which there
is a flipper who flipped that prefix." Under that interpretation: no.
>>>PROPOSITION 2
>>>"Regarding the infinite set of people flipping coins,
>>>the diagonal coin sequence has been flipped to an infinite amount of flips of the diagonal"
>>
>>False, though it is possible that every finite prefix of diagonal is one
>>of the prefixes of one of the flippers.
>
> You must ALWAYS be able to form a new sequence, (i.e. for ANY possible list of flippers),
> you have to consider that possibility.
>
> If all prefixes of the diagonal match prefixes of the flippers,
> then what portion of the diagonal is contained in all its prefixes? finite or infinite?
>
> What portion of the diagonal is matched on the list of flippers? finite or infinite?
Going for interpretations again:
1) "If every prefix of the diagonal is a prefix for some flipper, what
is the maximum length of the prefixes of the diagonal represented by the
flippers?" infinite
2) "If every prefix of the diagonal is a prefix for some flipper, what
portion of the diagonal is represented by a prefix?" finite
>>>PROPOSITION 3
>>>"Regarding the infinite set of people flipping coins,
>>>the diagonal coin sequence has been flipped some finite amount (or 0) of flips per person"
>>
>>True.
>
> The question is simple. You ASSUME a good random listing is achieved.
Wouldn't it be easier to just *state* that all computable listings are
present? Randomness is not part of logic, it's part of probability.
> Assume only a finite number of the diags flips occur in initial segments of the flippers outcomes.
> Then this segment has length L.
> There are 2^L possible initial segments of that length.
> Therefore, with oo flippers being infinitely larger than 2^L, it is improbable for each one
> of the 2^L initial segments not to have been flipped.
> Therefore, there is no such L.
>
> This case is ATLEAST one of the possibilites you have to consider to ALWAYS find a new sequence.
>
>
>
> THEN, THE QUESTION IS ABOUT THE NEW SEQUENCE.
>
> What portion of the new sequence are part of initial segments of the flippers?
>
> A FINITE PORTION
> INFINITE
Interpretations:
1) "What portion of diagonal has an initial segment in common with any
given flipper?" finite
2) "What is the upper bound on the length of an initial segment that
diagonal will have in common with some flipper?" probably no upper
bound, also called infinite.
--
Will Twentyman
email: wtwentyman at copper dot net
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