What is countably infinite set? [duplicate]

I am very confused with the definition of countable sets. What I ahev come to know about a countable set is, a countable set is a set of either a finite set or countably infinite set.ehat I understand from this is that a set having finite elements ( which we can count like 10, 50 etc) is called a finite set and hence its countable set but iam intimidated by the countably infinite set. Honestly I don't know what it means but what I understand is that it have many elemenys which are infinite but can be counted! And that is where I get confused. Like how we can count infinite elements. I know its pretty useless to ask here but I searched wiki but I was not able to understand. So any help will be admirable. Thanks

marked as duplicate by Mauro ALLEGRANZA, drhab, Jose Arnaldo Bebita-Dris, Namaste elementary-set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Sep 5 '18 at 13:30

• Yes; "countably infinite" means infinite but bijectable with the set $\mathbb N$ of natural numbers. – Mauro ALLEGRANZA Sep 5 '18 at 9:11
• A countable infinite set is a set where you can list the elements one-by-one, but your list is infinitely long. Some examples are the natural numbers, integers, and rationals. – Michael Burr Sep 5 '18 at 9:13

"Countable" may not be the best name to capture your intuition. Maybe thinking about the word "listable" instead works better for you. A set is countable / listable if you can, at least theoretically, write down a list of all the elements. The list is allowed to be infinitely long, but any spot on the list must be given a finite ordinal number: first, second, third, and so on.

The natural numbers is the canonical example of a countably infinite set. You can clearly make a list (albeit an infinitely long one) of all the natural numbers such that each element has its own, finitely numbered spot on the list.

There are infinities which are bigger, so that with any list, even infinitely long, you are guaranteed to miss at least one (and therefore, in fact, most) of the elements. The set of real numbers is a common example of this.

• I appreciate your view but if you could, then please Enlighten me about uncountable set – Noob Sep 5 '18 at 9:20
• I more thing thing I think set of real number is uncountable set? – Noob Sep 5 '18 at 9:24
• @low_burning That's a whole different question. There are loads of videos on youtube going through what's called "Cantor's diagonal argument" proving that there is no complete list of real numbers and that the set of real numbers is therefore uncountable, and you should look it up yourself first. The gist of the argument is that they take any proposed list, use that list to find a real number which is not on the list, point to that number, and say "You missed that one". Since it can be done with any list, any list is incomplete, and the real numbers are therefore not countable. – Arthur Sep 5 '18 at 9:25
• Why I think real numbers to be uncountable set Is That because, as said here that if a set have cardinality less than or equal to that of natural number then set is countable. So using this I say that take any two natural number say 1 and 2 now the cardinality of the this two is 2 (let's limit our discussion to these two only) now for these two we have many real numbers including 1, 2, 1.32,1.25...... And so on hence the cardinality of real number is greater than natural number. So they must be uncountable set. – Noob Sep 5 '18 at 9:31

A countably infinite set is a set $S$ for which exists some bijective map $f:\mathbb N \rightarrow S$ or the other way around. In other words, a set for which you can assign every member of the set a unique natural number, and "use up" all the natural numbers.

Using some basic theorems, you can switch the set of natural numbers with the set of integers, odd integers, even integers, rational numbers or any infinite subset of each of them.

One way to understand "countably infinite set":

You know that the set of natural numbers $\mathbb N$ is infinite - meaning it has an infinite numbers of elements. But for instance, you are still able to enumerate its elements: 0, 1, 2, 3, etc....

It works as well for $\mathbb Z$: you can count like this: -10, -9... 9, 10.

For $\mathbb Q$, it's a bit more tricky, but it can still work: $\mathbb Q$ is merely the set of all numbers which can be written $p \above 1pt q$ with $p \in \mathbb Z$ and $q \in \mathbb N$*. Since you can enumerate all elements of both $\mathbb Z$ and $\mathbb N$, you can enumerate all elements in $\mathbb Q$

It doesn't work for $\mathbb R$ - this is an "uncountably inifinte set".

If you try to list all the numbers in $\mathbb R$ between 0 and 1, you would start like:

0, 0.1, 0.01, 0.001, 0.0001... Without ever stopping or even reaching 0.2. You will always be able to find a number between 0 and 1 that you haven't listed yet. (see Cantor's diagonal argument)