Ambiguity for the countable behavior of a finite set

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?

• Did you roll a pair of dice when you chose the tags for this question? – Asaf Karagila Oct 18 '16 at 17:17

The term "countable" is often misused.

A set $X$ is said to be countable if it can be placed in 1-1 correspondence with some subset of the set of natural numbers.

A countable set can be either finite or countably infinite.

Some people use "countable" to mean "countably infinite," excluding finite sets. This usage makes no sense to me: a set should be countable if one can count or enumerate its members, which you can clearly do for a finite set. (I think this usage stems from situations where you're only talking about infinite sets, so that the distinction is, in that context, irrelevant.)

Does the definition of a natural number include $0$? Sure. Except when it doesn't.

In set theory, for example, it makes much more sense to include $0$ in the natural numbers, we can then say that a set is finite if and only if its cardinality is a natural number. It also makes it easier to define various things, without having to separate the empty case.

In analysis, on the other hand, it makes much more sense not to include $0$ in the natural numbers. Then we can write $\frac1n$ judiciously without worrying about $\frac10$ popping every time, or qualifying $n>0$.

Similarly, countable can include "finite" or it can exclude it. It all about what is easier to work with in a given context. Sometimes the finite case is an annoying difficulty which is not of interest, so countable is reduced to countably infinite; and sometimes it is the exact opposite in which case countable will include the finite sets.

In either case, however, "uncountable" always means infinite and not countable.

Just like everything else which is based on language, context is of utmost importance. And remembering that languages evolve to communicate ideas, often in the easiest and shortest way reasonable (while still conveying the details of the idea). Which is why "countable" can change its meaning, as can "natural number".

• The bad thing about preferring one definition over the other one in some context disrespects the later to some extent. – Nitin Uniyal Oct 19 '16 at 1:29
• However, you have shown something like having two wives at the same time with due respect to both of them. – Nitin Uniyal Oct 19 '16 at 1:33