 One thing this teaches us is that the probability axioms are silent on Hume’s problem. Inductive reasoning is compatible with the axioms, since Carnap’s way of constructing the prior probabilities makes a 10th \(\mathsf{T}\) quite likely given an initial string of \(9\) \(\mathsf{T}\)s. But the axioms are also compatible with skepticism about induction. On the first way of constructing the prior probabilities, a string of \(\mathsf{T}\)s never makes the next toss any more likely to be a \(\mathsf{T}\), no matter how long the string gets! In fact, there are further ways of constructing the prior probabilities that yield “anti-induction”, where the more \(\mathsf{T}\)s we observe, the less likely the next toss is to be a \(\mathsf{T}\).