Why set of natural numbers is infinite, while each natural number is finite? 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.


The fact that the infinitude of the natural numbers is effectively taken as an axiom in set theory illustrates to some degree the difficulty in proving that there are infinitely many natural numbers.  And you are correct to observe that "infinite" needs to be correctly defined before it makes sense to claim the natural numbers are infinite in number.
What we can quickly and easily do, is to prove that your claim there are finitely many natural numbers, is contradictory.
If as you state, $P(n)$ counts every natural number then since $P(n)=n+1$ it follows by the law of induction that $P(n)$ is itself a natural number and also by induction that $P(n)+1$ is a natural number.  Then the contradiction immediately follows that for every $P(n)$ there is some integer greater than it so every $P(n)$ fails to enumerate the natural numbers.
It does not take a huge leap of logic to add to our system the axiom that some number exists called $\infty$ which is greater than all natural numbers.
Intuitively, my preference is to a) aspire to only use infinity in supposition with a view to demonstrating a contradiction and b) also I prefer the notion that infinity is algebraically independent of addition rather than that it exists and is greater than every integer, but this is VERY FAR from standard.
I don't think it's wise to criticise a Fields medal winner for not providing a proof of an Informal Remark. You can rest assured he is extremely adept at constructing rigorous formal arguments when required to do so.
