Re: etc

David Marcus <DavidMarcus@xxxxxxxxxxxxxx> writes
Andy Smith wrote:
> David Marcus <DavidMarcus@xxxxxxxxxxxxxx> writes

>>The whole point of binary (or decimal) notation is that it represents a
>>real number. It would be kind of useless if it didn't. Have you taken
>>Calculus? Have you seen a proof that
>>
>> lim n->oo sum_{n=1}^oo a_n 2^{-n}
>>
>>exists (where each a_n is in {0,1})?
>>

To try to clarify my objection more, I would have said that

lim n->oo sum_{n=1}^oo a_n 2^{-n}

is not a number itself, but a representation of it (the infinite set
{a_n} defines one possible label for a unique real number).

That's nonsense. Try looking up the definition of limit in a calculus
book. You are confusing a name with the thing it names.

"lim n->oo sum_{n=1}^oo a_n 2^{-n}"

is the name of a real number.

lim n->oo sum_{n=1}^oo a_n 2^{-n}

is a real number.

With respect, I think you are confusing map with territory. That expression is just one of a number of possible identifiers for that number. e.g. if nothing else, base 3, base 4 base r etc representations.
The binary sequence is not the number, it is just a description of the point that you mean.

{a_n} can be defined by any rule, recursive or otherwise, provided that
it labels a unique real.

We prove that the limit exists for any sequence {a_n}. There is no
"provided".

Well if it identified more than one real, it wouldn't be much of a limit, would it? I had in mind possible definitions of the set of {a_n} that varied as a function of n (e.g. let a_i = 0, i= 0 to n, if n even, 1 otherwise), so that the number flip-flopped around and didn't have a well defined limit.

e.g. pi, e etc call all be specified by an
algorithm or formula that gives values for a_0 to a_n, for all n, to
give a number that can be proved to converge on pi or e etc in the
infinite limit. But you don't need to generate all the values in the
infinite set {a_n} for the definition of a_n to then stand as a
definition of (or label for) the unique real number.

No idea what "generate all the values" means.

You do not need to explicitly state or evaluate all the terms of e.g. pi = 3.141593625... for an algorithm/formula for a_n to stand as a label for pi.

But if you say, let a_n = RAND, where RAND is the output of a genuine
random number generator giving 0 or 1,

What is a "genuine random number generator"? Are we still doing math?

I meant one that was indeterminate i.e. not a pseudo-random number generator whose output was algorithmically determinate.

then this does not provide a
label for a unique real number because you need to generate the infinite
set of all a_n first to know what the real number is, and since {a_n}
has no last member you cannot specify a value for all n, and so there is
no limit n->oo that can be defined.

Let's go back to a simpler question, i.e., the one at the top of this
message. Forget the random numbers. I claim the following is a theorem.

Theorem. Suppose a: N -> {0,1}. Then lim n->oo sum_{n=1}^oo a(n) 2^{-n}
exists and is in [0,1].

Do you disagree?

No, provided that you can specify all of the members of the infinite set {a(n)}

Or am I confused as usual?

Extremely.

I thought I had just about got to grips with all this. Clearly not.

--
Andy Smith
.

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