On 03 Feb 2010, at 03:00, Jason Resch wrote:

On Thu, Dec 31, 2009 at 12:38 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:

UDA = Universal Dovetailer Argument. It is an argument which is supposed to show that if we take seriously the idea that "we" are digitally emulable, then we have to take seriously the idea that physics is a branch of number theory. Intensional number theory (number can serves as code for other numbers and functions: it is theoretical computer science, also).

Bruno, when you say code here, you are referring to code as in programming code, correct? I understand how a number can function as code for a function or a machine, but how can a number be code for another number?

Consider the first order arithmetic language in which the number 2 is denoted by s(s(0)). Now "s(s(0))" is itself a string, and when we translate meta-arithmetic in arithmetic (like Gödel) that string will be represented by some number like

2^'s' * 3^'(' * 5^'s' * 7^'0' * 11^')' * 13^')'

(Using some Gödel numbering). Then you can consider the function which send a number on its Gödel number. In this case it is more a cipher which is coded than a number per se (OK).

More simply a listing of telephone numbers. Each number entry of the listing can code a phone number. If you represent number by operator, like it is done with the combinators (where Church codes number n by the operator which iterates n times the input operator: [n(f)](x) = f(f(f(f(f(f ...f(x))))) n times. In that case number coding those operator will be code for the number represented by the operator.

Code, or index, program, machine are naturally defined by the phi_i, where i is the code of phi_i.

But I agree that coding number by number is not a good pedagogical idea. My idea was to remind that number can code anything capable of being described in a finite way like (partial) computable functions, and ... numbers.

You've said many times that all it takes for everything we see to exist are the natural numbers, addition and multiplication,

"to exist" in the sense of being apparent to us, not necessarily in the first order sense of existing, although we can collapse some form of existence through coding. For example, a finite piece of computation is an abstract object. But to give you an example of finite piece of computation, I will have to describe it by some finite object. But such a finite object should not be confused with the computation, even if it happens that we have, in that case (of finite piece of computations):

It exist a finite piece of computation going from state A to state B if and only if it exist a number coding that finite piece of computation.

This probably explains the confusion between computations and description of computation.

In step 8, it is important to understand that a movie of a computation is not a computation, but a description of a computation.

but where/how do functions and machines enter in to the picture? It is clear to me how once we get to the objective existence of functions,

... of computable functions. (Not all functions, unless we talk with a set theoretical Löbian machine like ZF).

we get the UDA, but I think I am missing some step.

It is nice you are aware of that. Keep hope, I have still to continue the seventh step serie. I am a bit stuck on how to explain the confusion mentioned above (between computation and description of computation, or even between number and description of number). It is a key for both the seven and the eight step.

Is your point that with addition, multiplication, and an infinite number of successive symbols, any computable function can be constructed?

You can say so.
You could also have said that with addition + multiplication axioms + logic, any computable function can be proved to exist.

Or do the relations imposed by addition and multiplication somehow create functions/machines?

You can say so but you need logic. Not just in the (meta) background, but made explicit in the axiom of the theory, or the program of the machine (theorem prover).


You are welcome. Such questions help to see where the difficulties remain. Keep asking if anything is unclear.



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