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 [itex]\mathbb{N}[/itex]. This is countably infinite by definition.

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

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

[tex]0\rightarrow 0,~1\rightarrow -1,~2\rightarrow 1,~3\rightarrow -2,...[/tex]

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 [itex]\mathbb{Q}[/itex] is countably infinite.

A set that is NOT countable infinite is [itex]\mathbb{R}[/itex].

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.

 

[blue_raver22]

Countably infinite refers to a set that has an infinite number of elements that can be counted, meaning that each element in the set can be assigned a unique number. This is in contrast to uncountably infinite sets, which have an infinite number of elements that cannot be counted.

For example, the set of all natural numbers (1, 2, 3, 4, ...) is countably infinite because each number can be counted and assigned a unique number. However, the set of all real numbers between 0 and 1 is uncountably infinite because there is an infinite number of numbers between 0 and 1 and they cannot be counted in a finite amount of time.

Another way to think about countably infinite sets is that they can be put into a one-to-one correspondence with the set of natural numbers. This means that for every element in the countably infinite set, there is a corresponding natural number and vice versa.

I hope this helps clarify the concept of countably infinite for you. If you have any further questions, please let me know.