I am asking this question to determine whether finite sets are countable? Or, my question is irrelevant as the term 'countability' is used for only infinite sets (though some authors have defined it for finite sets also). Even, some experts give different answers to this question with reference to two different definitions. But, I think that exactly one of them has to be correct.
My thought:: I believe that the definition,"A set $A$ is countable when a bijection $f:A→\mathbb N$ exists" is misleading. The reason being the truth value of the following statement:
If a set is uncountable then it is infinite.
Now, the contrapositive of the above statement is:
If a set is not infinite (or finite) then it is not uncountable (or countable).
Why the above definition is still in use if it is voilating a straight forward logical statement?