TITLE: Are professional mathematicians concerned with formalizing infinitely many dependent choices?
QUESTION [18 upvotes]: I've noticed certain arguments in analysis textbooks which rely on the principle of being able to pick elements infinitely many times. For example, an argument might go "Pick $x_1\in S$ such that $P(x_1)$. Having picked $x_1,x_2,\dots,x_k\in S$, pick $x_{k+1}$ such that $P(x_{k+1})$. This justifies the existence of a sequence in $S$ having properties X, Y, and Z." But this is not entirely induction; this is also recursion (and in some cases AC is used). And the recursion theorem/AC are totally absent from most textbooks. So how is this mathematical argument justified? Why doesn't the writer care about being explicit with the principles being used?
When I was an undergrad, this gave me anxiety.

REPLY [4 votes]: Mathamatics textbooks are written for humans. As such the author doesn't try to be explicit about everything that happens but assumes that the reader can follow certain arguments on his own.
On the other hand there's the field of automated theorem proving. In it proofs are formalized in a very explicit way because computers need explicit commands. That usually leads to proofs that are much more complex than the one in your math textbook.