Yeah I mean, I'm not claiming it has a big sense of obligation, only that it illustrates a condition where discourse seems to benefit from a sense of obligation.

Here's an example of a cheap question I just asked on twitter. Maybe Richard Hanania will find it cheap to answer too, but part of the reason I asked it was because I expect him to find it difficult to answer.

If he can't answer it, he will lose some status. That's probably good - if his position in the OP is genuine and well-informed, he should be able to answer it. The question is sort of "calling his bluff", checking that his implicitly promised reason actually exists.

Actually, we'll reschedule to make it for the meetup.

I didn't make it last time because my wife was coming home from a conference, and I probably can't make it next time because of a vacation in Iceland, but I will most likely come the time after that.

I don't have much experience with freedom of information requests, but I feel that when questions in online debates are hard to answer, it's often because they implicitly highlight problems with the positions that have been forwarded. For all I know, it could work similarly with freedom of information requests.

Ok, so this sounds like it talks about cardinality in the sense of 1 or 3, rather than in the sense of 2. I guess I default to 2 because it's more intuitive due to the transfer property, but maybe 1 or 3 are more desirable due to being mathematically richer.

Also the remark that hyperfinite can mean smaller than a nonstandard natural just seems false, where did you get that idea from?

When I look up the definition of hyperfinite, it's usually defined as being in bijection with the hypernaturals up to a (sometimes either standard or nonstandard, but given the context of your OP I assumed you mean only nonstandard) natural $n$. If the set is in bijection with the numbers up to $n$, then it would seem to have cardinality less than $n+1$^[1].

1. ^{^}

   Obviously this doesn't hold for transfinite sizes, but we're merely considering hyperfinite sizes, so it should hold there.

Hypernaturals are uncountable because they are bigger than all the nats and so can’t be counted.

This isn't the condition for countability. For instance, consider the ordering ${<}_{ext}$ of $N\cup \left\{\infty \right\}$ where when $n\in N$ then $n{<}_{ext}\infty$. This ordering has $\infty$ bigger than all the nats, but it's still countable because you have a bijection $N\to N\cup \left\{\infty \right\}$ given by $0\to \infty$, $n+1\to n$.

Also countability of the hypernaturals is a subtle concept because of the transfer principle. If you start with some model $M$ of set theory with natural numbers $N_M$ and use an ultrafilter to extend it to a model $M^{\ast }$ with natural numbers $N^{\ast }$, then you have three notions of countability of a set $S$:

1. $M$ contains a bijection between $S$ and $N_M$,
2. $M^{\ast }$ contains a bijection between $S$ and $N_{M^{\ast }}$ (which is equal to $N^{\ast }$),
3. The ambient set theory contains a bijection between $S$ and $N$.

Tautologically, the hypernaturals will be countable in the second sense, because it is simply seeking a bijection between the hypernaturals and themselves. I'm not sure whether they can be countable in the third sense, but if $N\subset N_M$^[1] then intuitively it seems to me that they won't be countable in the third sense, but the naturals won't be countable in the third sense either, so that doesn't necessarily seem like a problem or a natural thing to ask about.

Whether cardinality of continuum is equivalent to continuum hypothesis

Not sure what you mean here.

1. ^{^}

   Is it even possible for $N_M⊊N$? I'd think not because but I'm not 100% sure.

"Hyperfinite" is a term used in nonstandard analysis to refer to things that are larger than all standard natural numbers but smaller than a nonstandard natural number. It's not the same as uncountably infinite.

But yes, some uncountably infinite sets can be assigned a reasonable uniform probability distribution.

There's also another sense in which some uncountably infinite spaces can be smaller than some countable spaces, namely compactness.

I dunno, I feel like there's often a reason that there's considered to be obligations to generate answers. Like if someone pushes a claim on a topic with the justification that they've comprehensively studied the topic, you'd expect them to have a lot of knowledge, and thus be able to expand and clarify. And if someone pushes for a policy, you'd want that policy to be robust against foreseeable problems.

I can definitely see how there can be cases where there's an unreasonable symmetry in how questions vs answers can be valued compared to how expensive they are, but it seems wrong to entirely throw out the obligation to generate answers in all cases.