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.


Natural numbers can be arbitrarily large.
But arbitrarily large is not same as infinitely large.
Consider this infinite series- 0.9+0.09+0.009+0.0009+......
The sum of n terms of the above series = 1 - 1/10^n.
By the principal of mathematical induction it can be shown that 1-1/10^n will always be <1 for n belonging to the set of natural numbers.
It requires all the countably infinite terms of the above series for the sum to add to 1.Any natural number of terms sum would be < 1.
So clearly no natural number can be = infinity.
But mind you the sum is not 1 because we had put infinity in 1/10^n.
That operation is meaning less.Their are other formal methods to arrive at the result. 
