TITLE: noetherian induction
QUESTION [9 upvotes]: So I think I've misunderstood the principle of Noetherian induction as stated in the Hartshorne exercise II.3.16, or his statement is slightly incorrect. He says: "Let $X$ be a Noetherian topological space, and let $\mathscr{P}$ be a property of closed subset of $X$. Assume that for any closed subset $Y$ of $X$, if $\mathscr{P}$ holds for every proper closed subset of $Y$, then $\mathscr{P}$ holds for $Y$. (In particular, $\mathscr{P}$ must hold for the empty set.) Then $\mathscr{P}$ holds for $X$."
Why does $\mathscr{P}$ hold for the empty set? What if $\mathscr{P}$ is the property of being nonempty and $X = \varnothing$?

REPLY [3 votes]: I believe that the answer lies hidden within the depth of "vacuously true argument".
An argument of the form $\forall x\varphi$ is true if and only if there is no $x$ such that $\lnot\varphi(x)$.
An example I often used with my students was "If it I am drinking beer during the class then you are all elephants" (or something similar along these lines, usually including alcohol of some sort and a ridiculous entailment). It does not matter that I am talking to people, because I am now allowed to drink alcohol during class (as a teacher/TA anyway).
If we say that $P(x)$ holds for $x$ if for all $y<x$ holds $P(y)$. In this case $<$ is proper inclusion of closed subsets; since there are no proper subsets of the empty set, the argument $P(x)$ holds for $\varnothing$ vacuously, as above.