31. Hence, the halting problem is " undecidable " for Turing machines. 32. There are two distinct senses of the word " undecidable " in contemporary use. 33. It is even algorithmically undecidable whether a given finite presentation defines a nontrivial group. 34. :It's an undecidable problem, so no such algorithm can exist. 35. As with many undecidable questions, one can still attempt to give useful approximate solutions. 36. However the halting problem is provably undecidable and so such an algorithm does not exist. 37. An application of proof theory is to show that certain undecidable statements are not provable. 38. Later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. 39. However, type inference in System F ( without explicit type annotations ) is undecidable . 40. Rice's theorem says " any nontrivial property of partial recursive functions is undecidable "
