Turing Completeness
As you know, most widely used programming languages (especially imperative ones) are Turing-complete. And some - even relative to compile time, like, say, C++ with their templates.
And how is Turing completeness proved/disproved? In itself, this concept looks difficult to formalize.
2 answers
Formally, it is necessary to prove this from the point of view of the theory of recursive functions - a language is Turing-complete if and only if it allows you to write every computable function. In practice, it is usually enough to deftly appeal to [Church's thesis].(http://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis)
See. also "How to Prove a Programming Language is Turing Complete?".
For some reason, it seems to me that if we can implement a Turing machine in some language, then it means that it is Turing-complete. I could be wrong, though.