ionfish

It's pretty common in the UK these days.

Lots. For example, that every set has a cardinality (is bijective with some aleph). That's pretty intuitive, too, as (I think) are the following.

* Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other.

* Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X.

* Tychonoff's theorem: every product of compact topological spaces is compact.

Having written the following, I now wonder whether you meant something more specific by computation than I did, so I'm not certain whether my point is really a response. Could you spell out the details of your comment a bit more, and perhaps touch on the approach which Wolfram argues for?

* * *

Offloading computation only works when you understand what the computations are and why we do them. That's something that must be learned, it's not knowledge that springs fully formed into our minds as soon as we step into a classroom.

Carrying out computations thus gives us explicit and implicit knowledge of how the things we may eventually automate actually work. But it's also valuable because it trains us to compute in a precise and effective manner—a capability that remains useful later on. For instance, in logic it's often important to be able to carry out syntactic manipulations (e.g. into normal forms) in one's head, or even tacitly.

I'm sure there are plenty of examples from other areas of mathematics where computation is important, it's just that we do it so automatically that we don't think about it. Often I've found that students have trouble following proofs that take logically and computationally innocent steps without saying what's going on. Here I don't mean things like applying AC in the background, but just simple tricks like de Morgan's laws or taking the contrapositive. They have difficulty because they haven't taken those steps often enough themselves to have internalised them.

I don't care that nobody cares. I care that they pretend that they do. Fake friendliness is annoying, and good service is not formulaic.

Yes, it does rely implicitly on Cantor–Schröder–Bernstein. That might be a downside, but I think when working informally (that is to say, when not teaching a set theory course) one can simply assert that if there exist injective functions from sets A and B into one another then they are equinumerous.

That being said, although important in the theory of the order of the cardinal numbers, Cantor–Schröder–Bernstein doesn't show that the cardinals are totally ordered. That statement is actually equivalent to the Axiom of Choice, whereas as far as I'm aware Cantor–Schröder–Bernstein holds in ZF.

I think this becomes much more intuitive once one understands that the cardinality of any nondegenerate closed interval is the same as the cardinality of the continuum.