Countable and uncountable set I am reading about countable and uncountable set.
I am wondering how can we say that if a set is equivalent to N(one to one correspondence with natural number set) then it is countable? Before doing so we have to prove that N is countable.
I have searched a lot about this but not been able to find anything like this. So if anybody of you have something related to this please share.
May be my question appears utter nonsense to you but please bear with me.
 A: To say that a set is countable if it is in $1$-$1$ correspondence with a subset of $\Bbb N$ is the definition of the term 'countable'. Prior to a definition, a technical term doesn't have a meaning in mathematical context. From the definition it follows easily that $\Bbb N$ is countable, because it is in $1$-$1$ correspondence with itself via the identity map $n \mapsto n$.
Edit: We could make all the definitions we want, but the ones that stick are the ones that prove to be useful, and it has proven useful to have a special term for sets that are in $1$-$1$ correspondence with $\Bbb N$, more so than for other cardinalities. In set theory we can show that $\Bbb N$ is the 'smallest' infinite set in the sense that every infinite set has a subset that is in $1$-$1$ correspondence with $\Bbb N$. Also countable objects sometimes have peculiar properties, that differentiate them from similar objects of higher cardinalities.
A: $\mathbb{N}$ by definition is our reference set for determining countability.
The reason that we like to single out $\aleph_0 =|\mathbb{N}|$ as an interesting cardinality for sets stems from the way that we 'count' the number of elements of a set. Imagine that I give you a set $A$ and then I ask you to start counting the elements of the set. You start by saying one, two, three, four and so forth.. That means you map each element of the set to a natural number. If this counting stops somewhere, then your set is finite and you have defined an injective map $f: A \to \mathbb{N}$, the reason for injectivity is that you never would like to have more than one 'nth number', if it never stops, as you move on, you're exhausting the set of natural numbers and we can assume that after an infinite amount of time you've exhausted all natural numbers and mapped your set onto $\mathbb{N}$. That is the intuition behind why we take $\mathbb{N}$ as the reference, this is how we count numbers.
A: Note that every set can be put into one-to-one correspondence with itself. The identity function $\text{id}(x)=x$ is always a bijection between a set and itself.
In particular $\Bbb N$ has a bijection with itself, and so it is countable.
It might be worth pointing out that sometimes the definition of countable includes finite sets (sometime it doesn't). In that case we can define a countable set as a set which can be injected into $\Bbb N$. The identity function still works to prove that $\Bbb N$ is countable.
