The Sizes Of Infinity
I found that the concept of diffrent sizes of infinity maps quite well to some basic geometry, so I will try to teach the concept with some helpful examples via geometry.
Countable Infinity
Countable infinity is all positive integers. You could start counting a countable infinity but you would never stop counting.
This concept is easily maped to a 'ray'.
(This just looks like a fancy number line, the geometry will come later)
A ray has a starting point but no ending, just like how you could start counting a positive infinity but never reach the end.
Uncountable Infinity
Theres a couple types of uncountable infinity you can have. The first kind is comprised of all integers including negatives, you counld not start counting an uncountable infinity.
Uncountable infinity can be visualized with just a simple line
Uncountable Infinity (The other kind)
With the first kind of uncountable infinity you could pick a random number and start counting and make it to the next number but with this next kind of infinity you could truly never count even if you picked a random number. This kind of infinity contains all rational numbers.
This kind of infinity would have to be represented by a more complex line. Imagine an infinite line where you say that A is the midpoint of the line that means that both parts are equal to infinity/2 but that still equals infinity. So if we where to keep spliting each segment an infinite amount of times we would then have an infinite line with an infinite number of segments that are all infinitly long.