If that's intuitive for you already (great!) consider one step further: Rice's Theorem. Rice's Theorem in layman's term is "all nontrivial properties of a program are undecidable in the general case". "Undecidable" is the difficulty class of the halting problem (there are actually harder problems!).

A couple accessible examples:

1. Constant Propagation - we know that we'll never detect all compile-time constants because of Rice.

2. Exception frequency - I can write a function that has "raise new FooException()" in the code, but the function never raises, and you won't be able to prove it never raises.