Use Technical Terms Correctly
...I belong to the school that believes that functions and their values are sufficiently different that the distinction should be maintained.
“Sequence” means “function whose domain is the set of natural numbers.” When an author writes “the union of a sequence of measurable sets is measurable” he is guiding the reader’s attention to where it doesn’t belong. The theorem has nothing to do with the firstness of the first set, the secondness of the second, and so on; the sequence is irrelevant. The correct statement is that “the union of a countable set of measurable sets is measurable” (or, if a different emphasis is wanted, “the union of a countably infinite set of measurable sets is measurable”). The theorem that “the limit of a sequence of measurable functions is measurable” is a very different thing; there “sequence” is correctly used.
I have systematically and always, in spoken word and written, use “contain” for € and “include” for С. I don’t say that I have proved anything by this, but I can report that (a) it is very easy to get used to,
(b) it does no harm whatever, and (c) I don’t think that anybody ever noticed it.
Consistency, by the way, is a major virtue and its opposite is a cardinal sin in exposition....
...The best notation is no notation; whenever it is possible to avoid the use of complicated alphabetic apparatus, avoid it....
The rule of never leaving a free variable in a sentence, like many of the rules I am stating, is sometimes better to break than to obey. The sentence, after all, is an arbitrary unit, and if you want a free “/” dangling in one sentence so that you may refer to it in a later sentence in, say, the same paragraph, I don’t think you should necessarily be drummed out of the regiment. The rule is essentially sound, just the
same, and while it may be bent sometimes, it does not deserve to be shattered into smithereens....
Use Symbols Correctly
...How are we to read “€”: as the verb phrase “is in” or as the preposition “in”? Is it correct to say: “For x € A, we have x € B”, or “If x € A, then x € B”? I strongly prefer the latter (always read “€” as “is in”) and I doubly deplore the former (both usages occur in the same sentence). It’s easy to write and it’s easy to read “For x in A, we have x € B”; all dissonance and all even momentary ambiguity is avoided. The same is true for “С” even though the verbal translation is longer, and even more true for “ S?’. A sentence such as “Whenever a possible number is ^ 3, its square is ^ 9” is ugly.