And just verifying is better, but the point is the overall thrust of the type of argument is valid Bayesian evidence.
I think you are being way to glib about the possibility of analyzing these foundational issues with probability. But let’s take for granted that it makes sense—the strength of this “Bayesian evidence” is
P(ratio goes to 1 | PA is inconsistent) / P(ratio goes to 1)
Now, I have no idea what the numerator and denominator actually mean in this instance, but informally speaking it seems to me that they are about the same size.
We can replace those “events” by predictions that I’m more comfortable evaluating using Bayes, e.g. P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days) and P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days | Voevodsky will find an inconsistency in PA in the next 10 years). Those are definitely about the same size.
Sure. There’s an obvious problem with what probabilities mean and how we would even discuss things like Turing machines if PA is inconsistent. One could talk about some model of Turing machines in Robinson arithmetic or the like.
But yes, I agree that using conventional probability in this way is fraught with difficulty.
I think you are being way to glib about the possibility of analyzing these foundational issues with probability. But let’s take for granted that it makes sense—the strength of this “Bayesian evidence” is
P(ratio goes to 1 | PA is inconsistent) / P(ratio goes to 1)
Now, I have no idea what the numerator and denominator actually mean in this instance, but informally speaking it seems to me that they are about the same size.
We can replace those “events” by predictions that I’m more comfortable evaluating using Bayes, e.g. P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days) and P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days | Voevodsky will find an inconsistency in PA in the next 10 years). Those are definitely about the same size.
Sure. There’s an obvious problem with what probabilities mean and how we would even discuss things like Turing machines if PA is inconsistent. One could talk about some model of Turing machines in Robinson arithmetic or the like.
But yes, I agree that using conventional probability in this way is fraught with difficulty.