What is ∞ + ∞ = ????

[Kernel Sohcahtoa]

Countable infinite means that you can devise a scheme how to count through the set that will sooner or later reach any arbitrary element of it although.you never finish all of them. You can never finish counting through it because its infinite!

My bold. This is what confuses me. How is it countable if you can never finish counting them? I'm guessing it's a math term as I said, and a technical term as you say, that doesn't literally mean "countable" in the normal dictionary sense of the word.

, but for any element you choose beforehand you can be sure that it will be reached sooner or later. The simplest example is the integers. 0,1,2,3,4... this counting scheme never ends but will eventually reach any number you want even if you can never finish all of them. This is what countable infinite means.

I don't get it. But I don't know math terms like "element" and "integer" so that probably doesn't help.

As opposed to, say, all the real numbers between 0 and 1. There is no way to count through them one by one that ensures that every single one will eventually be reached.

This confuses me so much. I was starting to think maybe it means the real/natural numbers can be counted but not every number including ones with all decimals. But then you say it's opposed to all real numbers.

I don't get it.

I think unless I learn more math then you can keep trying to explain and I still won't get it hehe. Please forgive me if I'm being frustratingly dense.

Edit (the edit is in bold)

Hammy, before I proceed with this post, I want to make it clear that it is not my intent to be condescending: my intent is to try and be helpful to you.  With that said, I completely see the confusion in calling an infinite set countable: if a set is infinite, then why does it make sense to call it countable? I must admit that the concept of countable infinity in mathematics is difficult.  As a result, one of the most interesting things that I’ve found is that if I’m having trouble wrapping my head around a particular statement, then trying to find an equivalent statement to the one I’m having trouble with, might just enable me to appreciate/wrap my head around the concept. Thus, for me, I prefer to understand a countably infinite set by thinking of it in terms of having a 1-1 relation to the set of natural numbers ( I will elaborate on what I mean by 1-1 relation in my example in the next paragraph).  If we can show that an infinite set has a 1-1 relation to the set of natural numbers (the set containing the whole numbers 1,2,3,... up to infinity), then this means that our infinite set contains the same amount of elements as the set of natural numbers, and so, we can call our infinite set countable.


Now, I’d like to try and illustrate the 1-1 relation concept with the following example: suppose that, at an arcade establishment, we have a set containing all video game tokens, where the number 1 token is the smallest element, followed by the number 2 token, the number three token, and so on (continuing to infinity). Also, lets say  that there is an infinite set of video games to play (assume that all of these games are distinct).  Now, suppose that we have a particular arcade machine such that, whenever we insert any particular token into the arcade machine, the machine will always give us a distinct video game to play; and, each video game in our infinite set of video games can be traced back to a particular token that was inserted into our machine.  Hence, each game token that is inserted into our arcade machine will generate a distinct video game, and so, our arcade machine enables us to effectively map each game token in the infinite set of all game tokens to a distinct video game in the infinite set of video games to play (in theory, inserting all of the tokens will enable us to play every game in the infinite set), which is a 1-1 correspondence between those sets. 

In conclusion, in this example, the infinite set of game tokens is like the natural numbers, the infinite set of video games to play is like a countably infinite set, and our arcade machine is the clever way (a function in this case) that we can effectively establish a 1-1 relation between these two sets, which is equivalent to saying that our infinite set of video games to play contains the same amount of elements as the infinite set of game tokens (the natural numbers), and so, we can say that  our infinite set of video games is countable.  
 
P.S. I hope that this post was actually useful, and I really appreciate your inquisitiveness and curiosity in this thread.