Algorithms are presumed here to define functions over strings. The function represented by a string a is denoted as F[a]. This proof proceeds by reductio ad absurdum; we assume that there is a non-trivial property that is decided by an algorithm, and then show that it follows that we can decide the Halting problem, which is not possible, and therefore a contradiction.
Let us now assume that P(a) is an algorithm that decides some non-trivial property of F[a]. Without loss of generality we may assume that P(no-halt) = "no" with no-halt the representation of an algorithm that never halts. If this is not the case then this will hold for the negation of the property. Since P decides a non-trivial property it follows that there is a string b that represents an algorithm and it holds that P(b) = "yes". We can then define an algorithm H(a, i) as follows:
We can now show that H decides the Halting problem:
Since the Halting problem is known to be undecidable this is a contradiction and the assumption that there is an algorithm P(a) that decides a non-trivial property for the algorithm represented by a, must be false.