Tuesday, 26 April 2016

elementary set theory - proof: The set N of natural numbers is an infinite set



DEFINITION 1: A set S is finite with cardinality nN if there is a bijection from the set {0,1,...,n1} to S. A set is infinite if its not finite.



THEOREM 1: The set N of natural numbers is an infinite set.



Proof: Consider the injection f:NN defined as f(x)=3x. The range of f is a subset of the domain of f.




I understand that f(x)=3x is not surjective and thus not bijective since for example the range does not contain number 2. But what would happen if we were to define f:NN as f(x)=x? It is a bijective function. Doesn't that make the set of natural numbers finite according to the definition? What am I missing can somebody please tell me?


Answer



No. The definition of finite is f:{0,1,...,n1}S is bijective.



We know f:NN via f(n)=n is bijective, but this maps N onto N. It does not map {0,1,...,n1} onto N.



Basically, this prove N is finite if N is finite.


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