Refer to Turing's 1936 paper, page 248, last paragraph. I present the paragraph in verbatim below :
The expression "there is a general process for determining..." has been used throughout this section as equivalent to "there is a machine which will determine ... ". This usage can be justified if and only if we can justify our definition of "computable". For each of these "general process" problems can be expressed as a problem concerning a general process for determining whether a given integer n has a property G(n) [e.g. G{n) might mean "n is satisfactory" or "n is the Godel representation of a provable formula"], and this is equivalent to computing a number whose n-th. figure is 1 if G (n) is true and 0 if it is false.
Recall that in his paper, a machine is satisfactory iff machine never gets stuck. The method to determine if machine ever prints a symbol (0, 1) can be used to determine if machine is satisfactory, which is equivalent to determining if machine halts.
It has been stated in the paper that :
To each computable sequence there corresponds at least one description number, while to no description number does there correspond more than one computable sequence. The computable sequences and numbers are therefore enumerable.
With this I have no problem.
What I am unable to understand is why is the following true:
statement that a particular computer program halts is ultimately just a statement about integers [Scott Aaronson].
Observe that in the first paragraph, Turing has made essentially the same assertion.
All we did was encoding of machines using integers. Why does that allow us to say that property of machine is property of integers? Whether machine halts is a property of machine, not of integers.
So why are we saying, "each of these "general process" problems can be expressed as a problem concerning a general process for determining whether a given integer n has a property G(n) [e.g. G{n) might mean "n is satisfactory" or "n is the Godel representation of a provable formula" [Turing], and "statement that a particular computer program halts is ultimately just a statement about integers" ?