# Can all mathematical operations be encoded with a Turing Complete language? [closed]

In High School Computing I was taught the Structured Program Theorem - that you could implement any mathematical operation using:

1. Sequence
2. Selection
3. Iteration

After completing a Computer Science degree - we can express what is required for any computable function more formally with M-recursive functions:

1. The constant $0$ function
2. The successor function
3. Selecting parameters
4. Function composition
5. Primitive Recursion
6. The $\mu$-operator (look for the smallest $x$ such that...)

This being the minimal set of axioms. Translating this to code we get:

1. The constant 0
2. Incrementation _ + 1
3. Variable access x
4. Program/statement concatenation _; _
5. Countdown loops for ( x to 0 ) do _ end
6. While loops while ( x != 0 ) do _ end

But I've come away looking for proof. How do we know all this covers all computable functions? Is there an obvious branch of Mathematics for which this is not covered? Is there a shortcoming in Lambda Calculus where it is yet to cover off obscure Mathematical operations?

(Noting of course that

My question is: Can all mathematical operations be encoded with a Turing Complete language?

• By "mathematical operations" you actually mean "computable functions", right? – chi Jul 8 '16 at 14:44
• Well part of my question is 'what is a computable function and what are the limits of that'? So I went back to something more operational. – hawkeye Jul 8 '16 at 21:57