I was watching a tv show and they stated... If you know the exact position and velocity of every particle you can accuratly predict the future for the rest of time. However I know that you cannot know the exact position AND velocity of a sub atomic particle. How accurate is the first statement if it were possible?
In classical mechanics, and purely theoretical, this statement can be valid. Chaos theory ruins any attempt to make this real, however. And quantum mechanics tells us that "the exact position and velocity" do not even exist (it is not a problem of our limited knowledge!). If you know the wavefunction of every particle exactly, you can predict the future in deterministic interpretations of quantum mechanics (again, in theory, as chaos still ruins everything) - but even then the predictions (like many worlds) won't be like classical predictions.
If WHAT were possible? If you knew the exact position and velocity? If you could accurately [exactly] predict? As mfb has said, there is a problem with the concept. In no particular order, lets talk about some of them: Science works because most things obey continuity and differentiability relations. Meaning a small difference in initial state not only leads to a small difference in resulting state, but actually converges to smaller differences (by some measure) in resulting state. Chaos theory, shows that this is not true, that small differences will (in some very, very simple systems) diverge (to one or many or infinite numbers of very different resulting states). We know that the world is NOT described by classical nor by non-relativistic quantum physics. The equations of relativistic quantum field theory are not, in general, soluble exactly. The equations can only be solved for the most simple (symmetric) systems. Consider the three body problem. The solution is NOT analytic. Consider that most numbers are transcendental, implying that the computer's memory requirement to store "the exact" position (or velocity) of just a single particle is infinite, requiring all of the matter in the Universe to record it in order to proceed with the calculations. Then consider the number of particles, both real and virtual. Consider a REAL isolated system. To what degree is it actually isolated from the rest of the Universe? Is it shielded from black body radiation? from neutrinos? from gravity? No, it isn't...not even close. -=-=- The easiest answer is that the world is quantum and PROBABILISTIC, there are NO hidden variables which, if we knew them, would reduce the probabilities to 0 and 1. That is, for each instant of time, the probabilities multiply based on the previous instant's "exactitude". So, what is ∞ * ∞ * ∞ * ... ? How do you compute that? Your first statement was based on a "Classical Physics" world view which is known to NOT be correct.
My guess is if you could predict the future with 100% accuracy, by whatever method, that would force the wavefunction to collapse because you have knowledge. Without the wavefunction the universe would end and everything would vanish. It is for this reason the future cannot be known. You can't know what numbers are going to be drawn in the Lottery when they're using a quantum mechanical process, like radioactive decay, to pick the numbers.