# Re: HERC 97 SCI.MATH 0

**From:** ken quirici (*kquirici_at_yahoo.com*)

**Date:** 01/20/05

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Date: 20 Jan 2005 07:49:23 -0800

|-|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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