I have watched many videos on YouTube about it from people like vsauce, veratasium and others and even my math tutor a few years ago but still don’t understand.
Infinity is just infinity it doesn’t end so how can there be larger than that.
It’s like saying there are 4s greater than 4 which I don’t know what that means. If they both equal and are four how is one four larger.
In: Mathematics
Imagine an infinity that consists only of all the odd numbers : 1, 3, 5, 7, 9 etc.
Imagine an infinity that consists only of al the even numbers : 2, 4, 6, 8, 10 etc
Either of these groups contains only half of all the numbers, but is still infinite.
Now imagine an infinity that consists of ALL numbers : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc
That last group consists of twice as many numbers as either of the first two groups, and so is larger. But they are all infinite.
You can understand that there are infinite number of fractions between 0-1.
You can understand there are an infinite number of intervals between to subsequent cardinals.
So there is an infinite number of fractions an infinite number of times.
If after watching those videos you don’t understand, you’re probably better giving up anyway because they do a pretty darn good job at it, better than me.
I think the easiest way to think about this is to think of dimensions. Say you have a line that goes on to infinity. That’s one dimension of infinity. Now add another dimension to that. Thats infinity times infinity, which is larger.
Other folks are using integers (1, 2, 3, etc) and rational numbers (1.00000000000…1, 1.00000000000…2, etc), but my way of thinking about this is similar to dimensions, where the fractions of an integer are another dimension applied to those integers, so you have integers (infinite dimension) and the values between the integers (infinite dimension). Unlike simply adding a single dimension though, in this case we’re taking infinite infinities, because between each decimal is an infinity, and the set of decimals is also infinite.
There is a hotel that has an infinite number of rooms. A travel soccer team shows up, they are the team “infinite even” numbers. They take up every room. The next day team infinite odd shows up and needs rooms. The hotel isn’t big enough, all the rooms are filled. The hotels would need a larger infinite number of rooms to house both teams.
What about when team an infinite fractions shows up? Now we need a larger hotel with more rooms…
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