Is a compiler a kind of Gödel numbering program?
No. Consider the following two C functions:
int f(int a) {
    return a * 2;
}

int g(int a) {
    return a + a;
}

If we're being a bit liberal here, both are &quot;well-formed formula of some formal language&quot;. Yet most C compilers with optimization turned on will compile these two functions to the exact same code. This violates the uniqueness of a Gödel numbering.