No - the set of computable numbers does not form a
continuum. Continuity is related to the concept of limits: {x_i} is a
convergent sequence if
\forall \epsilon0, \exist N: |x_i-x_N|\epsilon.
A continuous space is one for which every convergent sequence
converges to a limit, ie
\exists x:
Joel:
It seems to me there is a great deal more information in PI than
just the 2 bytes it takes to convey it in an email message.
Russell:
Not much more. One could express pi by a short program - eg the
Wallis formula, that would be a few tens of bytes on most Turing
machines. Even
Joel Dobrzelewski wrote:
And please explain for me how this calculation involved
the continuum or infinite binary expansion of the symbol
pi in any meaningful way.
Sorry, missed getting in this riposte in the last post. What does a
binary expansion have to with the calculation .1 * 10 =
Joel Dobrzelewski wrote:
Ok, sorry for being a smart-ass. Instead of baiting the discussion
to make my point, I'll try to simply state the position clearly.
We humans cannot deal with infinite structures, like pi. Numbers
like pi and e and Omega and all the others are the devil! :)
Hello again Joel.
I think I can agree with you, in a pragmatic sense, with what you state
below.
I agree that any useful TOE should be able to be implemented on a (large
enough) computer. This computation can then SIMULATE the relevant or
important aspects of the universe we observe, or all
Joel Dobrzelewski wrote:
But I don't dispute this, as I wasn't talking about the finite
representation. I was talking about the infinite process / function that pi
represents.
Maybe this is obvious, but my whole point is that we are fooling ourselves
if we think we can compute physics
Obviously, what you're looking for is some kind of counter example. I
think the problem lies in not being able to determine at any point of
the calculation just how many digits of the limit you have found. For
the counterexample what we need is a computable series, which we know
converges, yet we
On 25-Jun-01, Russell Standish wrote:
Obviously, what you're looking for is some kind of counter example. I
think the problem lies in not being able to determine at any point of
the calculation just how many digits of the limit you have found.
OK, I can understand that. In order for a
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