Not exactly. Chaos theory generally states that, while we can theoretically exactly describe the behavior of a system using certain equations (e.g. Navier-Stokes equations), any infinitesimal change in the initial conditions of the system may have drastic effects on the outcome. The most common example is weather. Everything about the weather can be determined by solving the conservation equations for mass, momentum, and energy simultaneously. However, since we cannot accurately define the initial conditions or disturbances in the system (e.g. a butterfly flapping its wings, changing parts of the momentum and energy equations), we can only predict the outcome within certain limits. This is why multiple possibilities are shown for the path of a hurricane instead of a single, definitive path.
And since we can't determine the properties of every single particle in the universe (due to it being infinite and all), we can never come to a prediction that won't prove wrong after a certain amount of time. Even if we could, it would only work for inanimate objects and perhaps very low level lifeforms. We'd need to figure out how exactly the brain works before we could apply particle-based predictions to humans.
If there is some magic 'brainitron' (a particle in the brain that controls our thoughts 😛), then the equation could predict our behavior. However, as yet, the nature of 'consciousness' is undetermined in that it has not been linked to anything physical.
Since you nor I knows the physical/metaphysical origin of consciousness, to assume that it's linked to some physical process is simply begging the question.
The problem is not their number. Hell, by claiming that we can "predict anything", we're already assuming that we can determine the state of every single particle in the universe in an instant.
What he is saying is at best trivial and at worst confused. Imagine that the set of equations did not have to be comprehensible. Then we could have an infinite set of binary equations (quite a large infinity) each one corresponding to one aspect of the physical world (that is, to one aspect of sense perception). It is trivial then that such a set of equations exists. On the other hand if he means by "a set of equations" a finite mathematical forulation which tells us what happens in the universe then as time goes on we can approximate to a belief in the true set of equations by baysian updating on a prior with positive probability on each finite mathematical formulation. (More precisely if we restrict interest to the set of measurements in the past and future. Obviously things happened in the past which we do not know and so no amount of measuring in the present and future can tell us what happened then.)
This also happens in England, that natural scientists are given authority to speak out on any subject however ignorant they are about it. In this case Hawking mouths off on free will without understanding that choice does not imply free will. He has read in a book perhaps that free will conflicts with determinism (as it does) and then assumed that free will is the ability to choose, and then concluded that determinism is in conflict with the ability to choose. Then he says determinism means we can't choose anything and he is held up as an insightful thinker when all he has done is made a hash of issues that were quite well argued and even understood several centuries ago.
