Re: Computable functions/reals.

On Aug 18, 5:58 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
reaste...@xxxxxxxxx says...

Your function f(x) = 2x doesn't work, either,
even if we assume the input is a Cauchy
set of rationals. For example, say I give
you this sequence:

(9/10, 99/100, 999/1000, ...)

Your function returns:

{18/10, 198/100, 1998/1000, ...)

This hardly looks like a Cauchy sequence
for 2.000... to me.

Of course it is! What do you think a Cauchy sequence is?

I realized it was a Cauchy sequence right after I posted it.
Even so, I think this shows how useless Cauchy sequences
are for for doing calculations.

A Cauchy sequence of rationals just means convergent in
a mathematically precise sense. For a sequence q_1, q_2,...
to be a Cauchy sequence, it must be the case that for
any positive real number epsilon, there is a natural number n
such that the elements in the sequence with index greater than
n are less than epsilon away from each other. More precisely,

forall reals epsilon > 0,
exists natural n > 0,
forall naturals m > n,

|q_m - q_n| < epsilon

Basically, this means that if you go out to term number n
or greater, then any two terms differ by at most epsilon.

OK.
Isn't there something about a real number
is the limit of a Cauchy sequence?

How would I compute the limit of an arbitrary Cauchy
sequence given that I can read only a finite number of terms?

I am not sure you can compute if two arbitrary Cauchy
sequences are close to each other, let alone if they are equal.

I was looking at what happens when you multiply two
Cauchy sequences. For example, 1 times 1:

(9/10, 99/100, 999/1000, ...)
x
(7/8, 63/64, 511/512, ...)
=
(63/80, 6237/6400, 510489/512,000, ...)

I am guessing that I can keep doing this until I get
something really strange looking.

Are you claiming a Turing machine can determine
the limit of these types of sequences?

This is my challenge. I have no idea if this is possible.

Given two tapes, each with a Cauchy sequence written
as pairs of integers (like above), determine if the
two real numbers represented by these tapes differ
by more than 1/10.

I would also like to know if there is a limit on the
number of terms that have to be read from each tape.


Russell
- 2 many 2 count
.

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