
Re: say I flip a coin 100 times...
Posted:
Apr 18, 2005 12:35 PM


HERC777 wrote: >>probability that Herc's first n tosses match any of the >>first n sequences is zero, for all n, so the limit as n >>tends to infinity is also zero. > > >>In this case, Herc must form a new sequence, because we know >>his sequence is different in at least one place from each of > > > not only has this conclusion been proven wrong it is a non sequitur. > as n tends to infinity, the sequence is different at the nth place. > > 0.9.. is different to 0.9 at the 2nd place. > 0.9.. is different to 0.99 at the 3rd place. > > 0.9.. =/= 0.99...9 (n 9s) forall n. so the limit as n tends to infinity > is they are also unequal.
From this and other remarks you have made, I think we are using different definitions of being in an infinite list. Here are a couple of candidate definitions:
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Definition 1
If x is an element of a set X, and L=(l_0,l_1,...) is an ordered list of elements of X, then x is in L if, and only if, there exists a natural number n such that x=l_n.
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Definition 2
If x is an element of a set X, and L=(l_0,l_1,...) is an ordered list of elements of X, then x is in L if, and only if, at least one of the following two conditions is met:
Either
there exists a natural number n such that x=l_n
or
x is an infinite length sequence of symbols from some alphabet A, and for every natural number m greater than some natural number m0, if x_{1..m} is the length m prefix of x, there exists a natural number n such x_{1..m}=l_n
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If you agree with one of these definitions, please indicate which. If you agree with neither, please state your own.
Definitions matter. According to definition 1, 0.9.. is not in the list (0.9, 0.99, 0.999, ...). According to definition 2, it is.
Patricia

