# How does one define transcendental numbers (such as Pi) in theory of general recursive functions

On a turing machine and in the lambda calculus one can define transcendental numbers such as Pi, the golden ratio, etc. These are computatible functions with 0-arity that never terminate. In the theory of general recursive functions (which also defines the same class of functions as turing machines and lambda calculus) the output is always a natural number N, or undefined. How does one define transcendental numbers in the theory of general recursive functions (mathematical functions defined by Gödel)

## 1 Answer

If you are only interested in transcendental (or at least only irrational) numbers, you can safely identify the number $$x$$ with the function $$d_x : \mathbb{N} \to \mathbb{N}$$ that on input $$n$$ outputs the $$n$$-th digit of the decimal expansion of $$x$$.

If you want all real numbers, that does not lead to a satisfactory theory of computation. A suitable way to handle reals there is to let a function $$f : \mathbb{N} \to \mathbb{N}$$ code a real number $$x$$ if $$\forall n \in \mathbb{N} \ |\nu(f(n)) - x| < 2^{-n}$$ where $$\nu : \mathbb{N} \to \mathbb{Q}$$ is some standard bijection. This means that there are many functions coding the same real, but that is unavoidable.

• I understand that its not possible to define ALL reals [uncountable], and your answer really hits it on the head for me - Thank you.
– RFV
Commented Oct 10, 2022 at 17:02
• @RFV Thanks for pointing that out, I hadn't updated my profile in years.
– Arno
Commented Oct 10, 2022 at 17:07