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.

if Halts(x, x) then loop forever

You found a program that tells you if a program halts and your first fucking idea is using it to loop forever?

How about don't write programs that do this dumb shit? This proof is retarded. Post something more believable instead.