Maybe my question does not make sense because I lack understanding further, but I was curious to know if the arithmetic was complete in Turing?
As I understand it, a "calculation model" is a mechanism for calculating outputs from inputs. Thus, a "calculation" is simply a correspondence between inputs and outputs.
So, if say, the universe of possible inputs and outputs is: 1 and 2. These are all possible calculations:
1 -> 1 1 -> 2 1 -> 1,2 1 -> 2.1 2 -> 1 2 -> 2 2 -> 1.2 2 -> 2.1 1,2 -> 1 1,2 -> 2 1,2 -> 1,2 1.2 -> 2.1 2.1 -> 1 2.1 -> 2 2.1 -> 1.2 2.1 -> 2.1
Now, I think it's not even technically the complete game, because the full game would be infinite because I could have had repeated inputs and outputs like
1.1 -> 126.96.36.199. But at least that is the basic general that I understand.
And in my "computation model", I should be able to say, apply the X calculation to some inputs, where X is one of the mappings above and get the corresponding outputs.
So, from there, I understand that it is proven that the Turing model is able to map all the inputs to outputs on the universe of non-complex numbers.
My question is therefore: would arithmetic be Turing Complete in its ability to map inputs to outputs? Or are there correspondences that can not be formulated with the help of arithmetic, but can the Turing model do it?