How to prove finite sets are countable?

Question

This textbook definition seems to define "countable" to be "infinitely countable".

Are there any ways to use these definitions to prove that finite sets are countable?

Answer 1

No. Under these definitions, finite sets are not countable.

Answer 2

If $\sim$ means there is an injection from the left set into the right set, then $A$ finite does in fact imply $A$ is countable by this definition.

Answer 3

There are three approaches:

1) Some texts define "countable" as either finite or in one-to-one bijection to $\mathbb N$ (which this book calls $\mathbb J$). In which case you can't prove finite sets are countable because they are countable by definition.

2) Other text define $A$ being "countable" as there being an injection from $A$ to $\mathbb N$. In that case you can prove a finite set is countable.

3) THIS book defines "countable" as being in bijection to $\mathbb N$. In which case you can't prove finite sets are countable because by (this) definition they are not countable.

....

Definitions are definitions. They aren't universal truths. If you called a dog's tail a leg, it'd have five legs but it couldn't walk on all of them.