Halting problem = undecidability of all program properties?

[i]UNDERGRAD QUALITY[/i] incoming: An old thread on /math/ got me thinking about this. We know the typical proof uses
func G(x): if Halts(x, x) then loop forever
else false
and G(G) as a counterexample. Would this
func P(x): if ReturnsOdd(x, x) then 2
else 1
mean you can't prove a program always returns odd numbers because of P(P)? What about any other simple property like ``returns alphanumeric character'' or ``returns a valid s-expression''? You can always construct a function that ends in a contradiction by just returning the opposite thing.

I don't understand what you are trying to ask >>1-kun. The halting problem is whether a magical algorithm that can decide if any algorithm terminates or not for certain domain exist or not. And the answer is no, but of course you can create an algorithm that decides if some specific algorithm terminates or not. Read IATLC (Introduction to Automata Theory, Languages, and Computation).