In the proof of Theorem 2, you write “Clearly is convex.” This isn’t clear to me; could you explain what I am missing?
More specifically, let
) be the subset of obeying %20%3C%20b%20\%20\implies%20\%20\mathbb{P}\left(%20a%20%3C%20\mathbb{P}(\lceil%20\phi%20\rceil)%20%3C%20b%20\right)%20=1%20). So }%20X(\phi,a,b)). If ) were convex, then would be as well.But
) is not convex. Project ) onto in the coordinates corresponding to the sentences and %20%3C%20b). The image is %20\cup%20\left(%20%20[0,1]%20\times%20\{%201%20\}%20\right)%20\cup%20\left(%20[b,1]%20\times%20[0,1]%20\right)). This is not convex.Of course, the fact that the $X(\phi,a,b)$ are not convex doesn’t mean that their intersection isn’t, but it makes it non-obvious to me why the intersection should be convex. Thanks!
This is pretty close to how I remember the discussion in GEB. He has a good discussion of non-Euclidean geometry. He emphasizes that originally the negation of Parallel Postulate was viewed as absurd, but that now we can understand that the non-Euclidean axioms are perfectly reasonable statements which describe something other than plane geometry we are used to. Later he has a bit of a discussion of what a model of PA + NOT(CON(PA)) would look like. I remember finding it pretty confusing, and I didn’t really know what he was getting at until I red some actual logic theory textbooks. But he did get across the idea that the axioms would still describe something, but that something would be larger and stranger than the integers we think we know.