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 definition of

,

. So

which is true since for any set A, we have

.

Do you think its correct understanding ?