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
List every possible number between 1 and 2
1, 1.00001, …. 1.99999999 etc
Now do the same for all numbers between 10 and 20
Both are an infinite series, but the second is across a wider range.
This is not a perfect analogy, but as you can see two infinite series can have differing scales
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First letʼs try to define, what counting means. To count how much stuff is in a set you assign elements in the set to elements in another set. If two sets can be connected this way, that every element has exactly ine pair, we say that they have the same number of elements. If you can assign every element a number between 1 and 4 using every number only once, you have 4 elements. Every set that has 4 elements has the same number of elements.
What Cantor has proven, however is that you canʼt do this with natural and real numbers. No matter what system you use to assign real numbers to natural numbers, there would always be a real number that have no natural correspondent. Therefore these sets have different number of elements.
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Think of all even numbers. 2, 4, 6, 8… It goes on forever and is obviously infinite. Then think of all whole numbers: 1, 2, 3, 4… This also goes on forever and is obviously infinite, but at the same time it’s also clearly twice as big as there are twice as many numbers as in the set of even numbers.
Maybe if it helps, think of them both as infinitely large sets, but the natural numbers set is twice as dense as the even numbers set. And that’s why it’s considered bigger.
It’s about the parameters you set.
There are infinite even numbers and there are infinite numbers, but the second infinity is larger than the first.
Some infinite sets of numbers have a clear starting point and a clear way to progress through them, like the natural numbers (1,2,3,…). It’s very easy to count the numbers of this set, and therefore its size is said to be countably infinite.
Some infinite sets of numbers do not have a clear starting point and do not have a clear way to progress through them, like the real numbers. Take all the decimal numbers between 0 and 1. What number follows 0? 0.00000000…1? Not really. It’s impossible to count the numbers of this set, and therefore its size is said to be uncountably infinite.
One can use clever tricks, like [Cantor’s Diagonal Argument](https://en.m.wikipedia.org/wiki/Cantor’s_diagonal_argument ), to show that there are more real numbers than natural numbers, which is why we say uncountable infinity is larger than countable infinity.
Edit: The mathematically precise way to describe it is not to compare the size of “infinities”, but rather to compare the size of infinite sets. A mathematician would say that the size of the set of real numbers is larger than the size of the set of natural numbers.
Some great answers here, one thing I’d add is that infinity isn’t a number, it’s more of a concept. While we can get away with treating it like a number sometimes, we’ll eventually get to something nonsensical. For example consider
Infinity +1 = Infinity
Which seems pretty sensible right? If we subtract infinity as if it were a number we get
1 = 0
Which is obviously a load of rubbish. So thinking about infinity like a number that fits within our usual rules is the wrong thing to do
I think you’re kinda mixing things up. It’s not like there are bigger and smaller infinites.
It’s more than there are some sets that are infinite and some are still bigger than others.
For example, the set of positive numbers is the same size as the set of positive AND negative numbers combined. I’m sure you heard about that. However, the set of all continuous numbers between 0 and 1 is bigger than both of them combined
So from your question it seems like you are conceptualising infinity as a number. In this case, we want to be thinking about infinite sets, i.e collections of numbers following a definition that contain infinite members.
When comparing the size of infinite sets, we look for a bi directional function that can take any member from one set and map it to the other set.
For example, using the sets of all positive integers, and the set of all integers, we can come up with a formula that maps all even integers to the positive integers, and all odd integers to native integers in such a way that every item in each set is mapped to a single item in the other set. This means that the set of infinity are the same size.
If we now take the two sets of “all of the fractions between 0 and 1, and all numbers between 0 and 1”. We can map every fraction to a number between 0 and 1 by just writing it out as a decimal, but there are plenty of numbers that cannot be mapped to a fraction (i.e pi/3). So because every fraction has a corresponding element in all numbers, but not all numbers have a corresponding fraction, we can say the set of all numbers is bigger than the set of fractions, even though both are infinite
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