OP understands that, the quote says that the "the infinitude of real numbers is next largest to that of natural numbers", which makes it sound like it is next largest as in second largest as in less large as in lower cardinality. So it seems backwards. Maybe the quote means "next largest" as in larger.
I see. yeah I can see why it's confusing now -- we use "next-to-leading order" to mean a smaller effect than the leading order, but here, "next largest" means the one larger than natural numbers. Language is hard
Everything you say makes sense, with the exception of your expectation not to pay extra for an unusual order (for that cafe). Consider if I asked for my salad to be roasted, and balked at a surcharge on the grounds that they also have roasted brussel sprouts! I don't think it's up to you to decide what orders fit into their flow and which cost extra. I'm glad you found the places that will make the coffee the you like.
It’s not an unusual order, hot coffee is sold millions of times per day, and steaming comes with many coffee drinks, it’s par for the course with espresso drinks. Roasting a salad is weird, hot coffee is not, so your example is straw man. The “cold” in cold brew is not referring to serving temperature, that’s your own misunderstanding, so I find the suggestion that hot cold brew is weird to be pretty funny. As has been said many times in this thread, nobody balks at the idea of cold hot brew, nor do they charge extra.
Former barista, it _is_ an unusual order from the perspective behind the bar. Unusual in that I never have heard of someone ordering a warmed cold brew.
I'm not certain how I'd warm it up. I suppose it could be poured into a clean frothing pitcher and steamed directly, I'd somewhat worry that might dilute the flavor of the coffee.
I don't reckon someone's going to want me to microwave their cold brew, but it certainly seems like it'd be the quickest way to do it.
Most cafe workers get into flows of orders. Lattes means you always pump syrups into the cup, start the pour, then steam milk. Cold coffees usually means you ready the cup (syrups, milk) and pour the cold brewed coffee onto it.
Warming the cold brew totally breaks that flow, and is why it would be unexpected.
Hope that perspective helps. I do want to try it now though! I could imagine it being pretty good.
Coffee nerd, I like to play with this stuff.
Steaming cold brew (flash or regular) will give a very smooth frothy texture - almost nitro like foam. You can serve them over ice to get them back to cold, or serve warm.
Tasty if you use good coffee, and pretty unique honestly - "steamed iced americano" or "aerocano" are the two names I've heard if you want other people's reports on them.
You're right that it's very different from the experience of microwaved cold brew, and a customers response can be all over the place depending on what they're expecting.
I appreciate that, it does help, thank you. Indeed I found a few cafes that were used to it, and quite a few that weren’t. I think you’re totally right that this trips up some people’s flow especially when they’re not used to it. I’m okay with accepting it actually is unusual for some and just instead being the person complaining that it should be usual or expected even if it isn’t always. FWIW I have tried it microwaved, and it’s fine, but never in a cafe - baristas have always steamed it until warm or hot. That also works for me.
The biggest sticking point is not the heating of the coffee at all, it’s the widespread misunderstanding of the what the word cold means in the term “cold brew”. It is incorrectly assumed that brewed relatively cold (room temperature) means served ice cold. Somehow a lot of people can’t understand the verb brew has nothing to do with serving temperature. Does that make sense to you?
Although you are entirely correct in a technical sense, and it's common to serve hot brewed coffee cold, the opposite is so rarely desired that it will be considered 'weird' (in the sense of 'unusual') 9 out of 10 times.
Every extra process involved in making a coffee is going to add complexity and time to the workflow, which many cafes will elect to charge extra for.
There is some consternation in Australia about paying more for iced coffees compared to hot ones too:
I know, I know. You’re right. But… even if it is weird, it is so easy! It’s not really the default assumption that’s frustrating, it’s when I get push-back for a request that is normal and default for other espresso drinks, something trivially doable, something every barista does dozens and dozens of times a day.
In general, I agree with you, and for the record, I'm in favor of general restrictions or limitations on social media for children. At the same, I want to add that I was not allowed to watch violent cartoons or play violent video games as a kid in the 90s. I felt left out on the playground when I didn't know how to play X-Men or Power Rangers or whatever. But in retrospect it was fine. I'm not sure the social isolation factor in particular is as dire as you claim.
This is interesting. The study you cite and quote is about a transfer of money from "naive" credit card consumers to "sophisticated" credit card consumers, which correlates to "poor to rich", "less educated to more educated", etc. I'm even more interested in the transfer that occurs from both cash and non-reward-card consumers to specifically reward-card users.
I tend to think that this isn't inherent to institutions (or individuals), but rather that institutions that have this behavior tend to grow and last, whereas institutions that do not stay small and eventually disappear. A kind of survivorship bias.
In this case, it's not just sound vibrations but vibrations that you can feel in your house and bed! Truly difficult. Double or staggered stud walls really help.
I think it's much better because the teacher doesn't have to match guesses to students. For example, for each student the odds are 30/100, roughly one in three. And any duplicates can be matched by a single guess.
That formula is missing something because for over 100 students the teacher can't lose. (they get over 100 guesses and there are only 100 numbers possible)
More precisely, if there are N students, the probability is (min(N,100)/100)^N. This is 1 for N ≥ 100. And the probability at N=30 is indeed a tiny 2e-16, which shows that the children's "random" picks were far from uniformly random.
(Incidentally, even with N=99 the probability is 0.37 ≈ 1/e, and the probability is lowest at N=37 ≈ 100/e. This is not a coincidence.)
The teacher picks 30 numbers out of 100. Then each student (independently) picks one number. If random, that is 0.3 ^ 30. Obviously, the students are not picking random.
