I am doing the chapter "Equinemerous sets" from Velleman's "How to prove it" and
I have some doubts. At one point he says that "for each natural number n , let
. A set
is called if there is a natural number n such that
. Otherwise A is . "
Further down he says that "it makes sense to define the number of elements of
a finite set A to be the unique n such that . This number
is also sometimes called the cardinality of A and its denoted
. Note that according to this definition,
is finite and ."
So that will mean that we will need to choose n=0 for an empty set, so that
. Now according to the author's defnition of
, . So
which is true since for any set A, we have .
Do you think its correct understanding ?