You do need more bandwidth. Trivial counterexample (for any reasonable inference of the semantics of the source code of this made-up language):

  define isCorrectlyTyped( program P)
  as
    return "no".

By the way: compilers for languages such as Java and C# do something similar: they reject all programs that may use a variable before a value is assigned to it, at the price of rejecting some programs that properly assign a value to each variable before use on the grounds that the compiler cannot prove it. Typically, the language specification describes exactly how smart compilers can be (or rather, how stupid compilers must be) when checking this, to ensure that all compilers agree about what programs are valid.