This question is rather short, but I couldn't find the specific answer I was looking for anywhere.
My understanding so far: (Please correct me if I am wrong on some of this)
- I can assume any statement $P(x)$ for all $x \in \emptyset$ to be true because the empty set doesn't contain any elements in the first place.
- I can also assume any statement $Q(x)$ to be false for all $x \in \emptyset$ because of the same reason.
- An example would be $ \forall M \subseteq \emptyset $ where $M \neq \emptyset$, it holds that $M = \emptyset$ . This statement has to be true for all subsets $M$ because the premise (there are no subsets of $\emptyset$ which are not the empty set themselves) is false.
My question: What is this good for? Is this behavior just a trivial property of the empty set? The only usage that comes to my mind is if I want to prove something specific for the empty set itself (for example that $\emptyset$ is an ordinal).