Re: Cantor Diagonal Proof

On 06/04/2025 07:43, Lawrence D'Oliveiro wrote:

On Sun, 6 Apr 2025 07:27:43 +0100, Richard Heathfield wrote:
 

On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
>

On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
>

But to be computable, numbers must be computed in a finite number of
steps.

>
“Computable Number: A number which can be computed to any number of
digits desired by a Turing machine.”
>
<https://mathworld.wolfram.com/ComputableNumber.html>

>
"The “computable” numbers may be described briefly as the real numbers
whose expressions as a decimal are calculable by finite means." - Alan
Turing.
>
And therefore, to be computable, numbers must be computed in a finite
number of steps.

 I would say you are quoting Turing out of context.

There is no context before his words because they are the paper's opening words. Here's the context that immediately follows.
"The “computable” numbers may be described briefly as the real
numbers whose expressions as a decimal are calculable by finite
means. Although the subject of this paper is ostensibly the
computable numbers, it is almost equally easy to define and
investigate computable functions of an integral variable or a
real or computable variable, computable predicates, and so forth.
The fundamental problems involved are, however, the same in each
case, and I have chosen the computable numbers for explicit
treatment as involving the least cumbrous technique. I hope
shortly to give an account of the relations of the computable
numbers, functions, and so forth to one another. This will
include a development of the theory of functions of a real
variable expressed in terms of computable numbers. According to
my definition, a number is computable if its decimal can be
written down by a machine." - Alan Turing

By your
(mis)interpretation of his words, even something like 1/3 is an
incomputable number, since its “expressions as a decimal are not
calculable by finite means”.

"If a computing machine never writes down more than a finite number of symbols of the first kind it will be called circular. Otherwise it is said to be circle-free." - Alan Turing.
"A sequence is said to be computable if it can be computed by a circle-free machine." - Alan Turing.

Turing would not have been so dumb as to have his definition of
computability depend on something as trivial as the choice of number base.

I'll let you read the paper yourself. It's not hard to find.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
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