Mathematically speaking, pages 2-3 appear sound (if one assumes the Axiom of Choice, the validity of which I can see being a big if for philosophers). The set W_f indeed has measure 0.
This already striking result can be made even more startling. In predicting the value of a function f at t, one does not actually need to know the behavior of f at all points less than t. In fact, no particular values of f need to be known. To predict f (t), all that an appropriate rule would require are the values that f takes on in the interval (t – ε, t), for any ε > 0.
This part seems to contradict itself. No particular values are needed but the values of f on the interval are required? That being said, the bolded part in the statement is obvious for continuous functions, but in the context of general functions, I find it quite surprising.
Since the paper by Hardin and Taylor is forthcoming, I will reserve full judgement until I have seen the proofs of some of the other statements he makes (the one in footnote 3, particulary and I'd have to see a proof of footnote 5.)
Now, whether this can be used to solve the problem of induction ... in classical mathematics, we make heavy use of the Axiom of Choice, proofs by contradiction, principle of the excluded middle, etc. In some sense, these appear to be needed of any theory that will attempt to explain the "world" around us (i.e. a statement is either true or false, I can always chose a sock from a pair of socks, etc...)
But in some wider sense, I wouldn't trust them worth of ****. It is not
at all clear that the Principle of the Excluded Middle and the Axiom of Choice hold in the Real World. Since the proof makes heavy use of EM and AC, I am still un-convinced that this beautiful piece of math can be used in the philosophical setting. But I don't know what constitutes a valid philosophical argument, so it might be all the rage by the time you respond to the post.