tl;dr: Every Turing Complete system is undecidable - i.e. for every Turing Complete system which answers "yes" or "no", you can't determine in finite time wheth

tl;dr: Every Turing Complete system is undecidable - i.e. for every Turing Complete system which answers "yes" or "no", you can't determine in finite time whether, for any given input, the program will accept it or not. Long version: Being Turing Complete means that a system can compute everything that a Turing Machine can compute. For a long time, it was an open question whether there was a way to determine whether, for a given statement, one could determine whether it was universally valid. Alan Turing's creation of the Turing Machine was originally in order to try to answer this question. The idea was that to determine if a statement was universally valid, one would need to be able to mechanically derive the statement from the axioms of logic - i.e., there would be an algorithm (which Turing formalized as a "Turing Machine") that would be able to take the input and decide whether it was universally valid. Turing then proved - likely his greatest claim to fame - that such an algorithm could not exist - as one can always build an input for which it will be forced to give the wrong answer. Hence, Turing showed that for any Turing Machine - or any system equivalent to a Turing Machine - which answers "yes" or "no" to some input, it is impossible to decide whether it accepts its input in finite time.