Originally Posted by

**issacnewton** At one point he says that "for each natural number n , let

$\displaystyle I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$. A set $\displaystyle A$

is called $\displaystyle \mathit{finite}$ if there is a natural number n such that

$\displaystyle I_n \sim A$ . Otherwise A is $\displaystyle \mathit{infinite}$. "

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 $\displaystyle I_n \sim A$. This number

is also sometimes called the cardinality of A and its denoted

$\displaystyle \lvert A \rvert$. Note that according to this definition,

$\displaystyle \varnothing$ is finite and $\displaystyle \lvert \varnothing \rvert =0$."

So that will mean that we will need to choose n=0 for an empty set, so that

$\displaystyle I_0 \sim \varnothing$. Now according to the author's definition of

$\displaystyle I_n$ , $\displaystyle I_0=\varnothing$. So $\displaystyle I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$ which is true since for any set A, we have $\displaystyle A \sim A$.

Do you think its correct understanding ?