The idea of the proof is simple: assume we have a prediction device which answers "yes or no" questions, then show that there is a Boolean function of the state of the Universe that the prediction device cannot predict.
What is this function? It is the negation of the answer given by the prediction device! This is considered a function of the state of the universe because the prediction device exists within the universe. Stripped of all its complex notation and jargon, the proof is a restatement of a logic puzzle: if there is a computer that gives "yes or no" answers and knows everything, what question can you ask that it will not be able to answer?
"Will your answer to this question be 'no'?"
Does this really refute determinism?
Wolpert's model of a prediction device is defined as a pair $C$ of functions $(X,Y)$ with domain the possible wordlines (in philosophy jargon, nomologically possible worlds) $u$ of the Universe $U$, where $X$ is the 'setup function' (with no codomain defined in the paper) and $Y$ is the 'answer functon' with codomain $\lbrace 0,1 \rbrace$. The setup function $X$ maps to initial states of the prediction device--$X(u)$ is the initial state of the prediction device in the worldline $u$ of the Universe, including its input.
Then the proof simply shows that the device $C = (X,Y)$ cannot predict the function $\neg Y$: whenever $Y(u)=1$, $\neg Y(u)=0$, and obversely.
Note that there is no dependence of the answer function on the setup function. For distinct answer functions $Y, Y'$, prediction devices with identical setups $C=(X,Y)$ and $C'=(X,Y')$ give different answers. To prove non-determinism, Wolpert has assumed non-determisim!
This has the strange consequence that if we built a machine $C = (X,Y)$, and found that there were a function of the universe $\neg Y$ that our machine could not predict, there would be a machine $C' = (X,\neg Y)$, physically indistinguishable in its initial state from $C$ in all possible worlds (since the setup function is the same) which predicted the "unpredictable" function $\neg Y$ perfectly!
To remove this circularity, could we use this argument as a reductio, i.e., assume determinism, so that any machines with distinct answer functions $Y, Y'$ must have distinct setup functions, and then derive a contradiction from the assumption that some machine predicts the negation of its answer function? No: the argument becomes completely trivial--of course the machine $(X,Y)$ doesn't predict the function $\neg Y$, because our assumption of determinism implies that for each $u$, the initial state $X(u)$ determines the output $Y(u)$.
In other words, if $Y$ is not determined by $X$, then although it is trivial that a machine with answer function Y cannot predict the function ~Y, we can at least claim that "No matter how the device is set up, there is a function of the state of the universe it cannot predict"--as Wolpert says in the paper, even if the device is given the correct answer in its input, it cannot predict the output of the value of the function correctly. This sounds impressive, until we realize that this is because we've already assumed non-determinism.
But if we assume that the initial state of the device determines its output, then the proof result reduces to "If a machine is set up to output some values, then the values it is set up to output are not equal to the negation of these values."
We have either circularity or triviality.
The door to determinism remains open.
