When I was younger...
The fact is surprising when coupled with the fact that particles do not have a definite spin direction before you measure it. The anti-correlation is maintained non-locally, but the directions are decided by the experiment.
A better example is: take two spheres, send them far away, then make one sphere spin in any orientation that you want. How much would you be surprised to learn that the other sphere spins with the same axis in the opposite directions?
How probable is that someone knows their internal belief structure? How probable is that someone who knows their internal belief structure tells you that truthfully instead of using a self-serving lie?
The causation order in the scenario is important. If the mother is instantly killed by the truck, then she cannot feel any sense of pleasure after the fact. But if you want to say that the mother feels the pleasure during the attempt or before, then I would say that the word "pleasure" here is assuming the meaning of "motivation", and the points raised by Viliam in another comment are valid, it becomes just a play on words, devoid of intrinsic content.
So far, Bayesian probability has been extended to infinite sets only as a limit of continuous transfinite functions. So I'm not quite sure of the official answer to that question.
On the other hand, what I know is that even common measure theory cannot talk about the probability of a singleton if the support is continuous: no sigma-algebra on 2ℵ0 supports the atomic elements.
And if you're willing to bite the bullet, and define such an algebra through the use of a measurable cardinal, you end up with an ultrafilter that allows you to define infinitesimal quantities
Under the paradigm of probability as extended logic, it is wrong to distinguish between empirical and demonstrative reasoning, since classical logic is just the limit of Bayesian probability with probabilities 0 and 1.
Besides that, category theory was born more than 70 years ago! Sure, very young compared to other disciplines, but not *so* young. Also, the work of Lawvere (the first to connect categories and logic) began in the 70's, so it dates at least forty years back.
That said, I'm not saying that category theory cannot in principle be used to reason about reasoning (the effective topos is a wonderful piece of machinery), it just cannot say that much right now about Bayesian reasoning
Yeah, my point is that they aren't truth values per se, not intuitionistic or linear or MVs or anything else
I've also dabbled into the matter, and I have two observation:
The way it's used in the set theory textbooks I've read is usually this:
So, there is indeed the assumption of a kind of infinite process before the assumption of the existence of an infinite set, but it's not (necessarily) the ordinal ω. You can't also use it to deduce anything else, you still need Replacement. The same can be said for the existence and uniqueness of the empty set, which can be deduced from the axioms of Separation.
This approach is not equivalent nor weaker to having fiat transfinite recursion , it's the only correct way if you want to make the least amount of new assumptions.
Anyway, as far as I can tell, having a well defined theory of sets is crucial to the definitions of surreals, since they are based on set operations and ontology, and use infinite sets of every kind.
On the other hand, I don't understand your problem with the impredicativity of the definitions of the surreals. These are often resolved into recursive definitions and since ZF-sets are well-founded, you never run into any problem.
> Transfinite induction does feel a bit icky in that finite prooflines you outline a process that has infinitely many steps. But as limits have a similar kind of thing going on I don't know whether it is any ickier.
Well, transfinite induction / recursions is reduced to (at least in ZF set theory) the existence of an infinite set and the Replacement axioms (a class function on a set is a set). I suspect you don't trust the latter.
The first link in the article is broken...