In his book Analysis Vol. 1, author Terence Tao argues that while each natural number is finite, the set of natural numbers is infinite (though has not defined what infinite means yet). Using Peano Axiom, if a property holds for P(0) and whenever P(n) is true, P(n+1) is also true, then it is true for all natural numbers. [See image attached at the end.]
However, he has not provided an argument/proof why the set of natural numbers in infinite.
If we go by the same argument, the set of natural numbers should also be finite.
Just like finiteness let’s say P is property called count associated with each natural number. Count can be defined as P(n) = n+1. (Intuitively count means number of elements in the set till n, or the number of elements in the set till n). Now P(0) = 1, which is finite. If P(n) is finite (i.e. n+1), then P(n+1) will also be finite. Hence, the number of elements in the set of natural numbers should also be finite.
Best Answer
Suppose for the sake of contradiction that the set of natural numbers is finite. Then there exists a maximum element $m$. But $m+1=n$ is also a natural number and $n>m$. This contradicts the maximality of $m$, so our original assumption was false, and hence set of natural numbers is not finite but rather infinite.
The induction argument fails because it shows $P(n)$ is finite for every natural number $n$, but it does not show that $P(\infty)$ is finite because $\infty$ is not a natural number.