What do you mean by countably infinite ?

[iVenky]

What do you mean by "countably infinite"?

I just couldn't understand the meaning of countably infinite. I have seen some definitions but I couldn't get an insight. Could you please help me in understanding this term with some kind of an example?

Thanks a lot.

:)

 

[micromass]

The most important example is $\mathbb{N}$. This is countably infinite by definition.

Furthermore, if there exists a bijection between $\mathbb{N}$ and a set X, then that set X is also called countably infinite.

As further examples, $\mathbb{Z}$ is countably infinite as there exists a bijection between $\mathbb{Z}$ and $\mathbb{N}$. The bijection in question is

$$0\rightarrow 0,~1\rightarrow -1,~2\rightarrow 1,~3\rightarrow -2,...$$

So you send an even number 2n to n, and you send an odd number 2n+1 to -n-1.

Another example is the set of even numbers. This is also countably infinite. The bijection sends n to 2n. So 0 is sent to 0, 1 to 2, 2 to 4, 3 to 6, etc.

A little harder to prove is that $\mathbb{Q}$ is countably infinite.

A set that is NOT countable infinite is $\mathbb{R}$.

Read this FAQ: https://www.physicsforums.com/showthread.php?t=507003

 

[iVenky]

So "countably infinite" means you can count like one, two, three so on.
You can't count real numbers between 0 and 1 like one, two, three.. so it should be an infinite space and not countably infinite.

Am I right?

 

[micromass]

So "countably infinite" means you can count like one, two, three so on.
You can't count real numbers between 0 and 1 like one, two, three.. so it should be an infinite space and not countably infinite.

Am I right?

Yes, the reals are uncountable infinite. You can't label them one, two, three, four, etc. and expect to have them all.
The rigorous proof that the reals are uncountable uses Cantor's diagonalization and is a really neat trick in mathematics.