Some Fundamental Theorems in Mathematics (2019) [pdf]

Based on the username and its recent birth, you created this account just to paste this comment. What does it mean? (I think that you are commenting sarcastically on the way math is taught, but that's just a guess.)

Oops, thanks! Didn't try "oe".

It's great to see someone assemble some of their favourite theorems, but, if you have occasion to use these, make sure that you do so with caution.

Aside from some typos ('Perseval' for 'Parseval') and some curiosities (stating on p. 3 that the cardinalities of a set and its power set are different, which is true; but why not say that the latter is larger?), I noticed a wrong statement on p.7: assuming that to 'extend' a monoid to a group is to realise it as a submonoid of a larger monoid that is also a group, this cannot be done in general. (For example, the 'min-monoid' whose objects are the natural numbers, and in which the product of two natural numbers is their minimum, does not embed in a group because it obeys no cancellative law.) What is true is that, for every commutative monoid, there is a universal homomorphism from that monoid to a group, in the sense that all other such homomorphisms factor through it. (In my example, this universal homomorphism is the constant-0 homomorphism to the 0 group.)

How useful is this? Not much that these things have in common beyond "results cited a lot".

Can't say that the subjects are chosen very precisely either -- the Fundamental Theorem of Algebra isn't actually a theorem of algebra; Tychonoff's theorem is fundamental only to the set-theoretical part of topology; the Fundamental Theorem of Counting is just a particular case of the "Fubini" interchange-of-summations formula, which I would call the real fundamental theorem of counting. The number of platonic solids is mostly a curiosity from a modern perspective; so are the transcendences of pi and e.

Also, the word "extended" in the Number systems section needs to be taken with some artistic license; the "extension" introduces new symbols (negatives) and new relations that can occasionally render some old elements identical (in the worst case, the whole monoid can collapse to a point). The most famous example of this is what happens if you divide by 0 (= extend the multiplicative monoid of real numbers to a group). This is not something the author should be blamed for; it's the only real error I've spotted at a quick skim, and it's rare for a collection as diverse as this to have this few errors. The real problem is: what's the point of such a survey if pretty much any of the results is given so little time and space that only those who already know it can understand it from the description?

Great point. There are a ton of good models that break down in certain regions (e.g. quantum mechanics and general relativity at large and small physical scales, respectively).