TITLE: Can all mathematical operations be encoded with a Turing Complete language?
QUESTION [6 upvotes]: In High School Computing I was taught the Structured Program Theorem - that you could implement any mathematical operation using:

Sequence
Selection
Iteration

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

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

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

The constant 0
Incrementation _ + 1
Variable access x
Program/statement concatenation _; _
Countdown loops for ( x to 0 ) do _ end
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 

Lambda Calculus is a subset of Term rewriting, 
that Mathematica chose Term-writing over Lambda Calculus as its foundation,    
and recent work in Univalent Foundations. )

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

REPLY [2 votes]: But I've come away looking for proof. How do we know all this covers all 
  computable functions? 

The Church-Turing hypothesis is a belief. So the common used term "computable function" is mostly that what can be computed in these models of computation.
As far as I understand it, at the heart of the classical computability models there is a mismatch in cardinality:
We have countable many programs (be it finite strings, or natural numbers) and uncountable many machine functions (the assignments between possible input and output of a machine). 
If you decide for a particular computation model, you will assign part of the machine functions to the programs. Due to the mismatch in cardinality, there are machine functions left over which do not get a program assigned, these are the uncomputable functions.
The classical models of computation deliver the same computable functions, because they share certain properties.

Is there an obvious branch of Mathematics for which this is not covered?
  there a shortcoming in Lambda Calculus where it is yet to cover off
  Mathematical operations?

E.g. there is a branch of analysis, computable analysis, which has its own
model of computation. I remember vaguely that multiplication of any real number by 3 is not possible in that model.
There is also a connection between constructivist mathematics and computation theory. 
Maybe this question will attract an expert who can
elaborate on this.