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.


I am reading the book and I had the same query. I think Tao's presentation of the idea is incorrect. I say that because the property of a natural number being finite cannot be deduced from the axioms Tao's provided. This is so for the axioms he's provided do not address the finite property of all natural numbers. In addition, his induction argument is unnecessarily confusing because one has to assume that 0 is finite and its successors are also finite leading to a circular argument. 
I think Tao's made an error in the way he's presented the idea. I think he wants to say that a set that consists of only one natural number is finite and the set containing all natural numbers is infinite. The reason I say his argument is circular is because the finiteness or infiniteness of a set is dependent on the definition of counting (mapping the elements of a set to a set of natural numbers). In the chapter that Manish is referring to, Tao wanted to present an idea on the inductive property of natural numbers and made a mistake in choosing the example he's chosen (since the example depends on definitions he hasn't presented yet). 
