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
So you can count: 1, 2, 3… and so on to infinity
Or you can count the even numbers: 2, 4, 6.. to infinity.
Which one is larger(has more numbers)? Counting normally.
In your day-to-day life you are mostly confronted with normal numbers and by that I mean so called “real numbers” (reals). There is a way to determine if one real is larger than another real – this way doesn’t apply to infinity, because infinity isn’t a real number.
For example the length of a stick can be modelled as a real. A stick has a “greater” length than another stick, when you lay them next to each other and the longer stick goes on, while the shorter stick already stopped. long – short > 0
For numbers or number-like things that include the real numbers, you need a different way of thinking about the “greater-than” relation, that nevertheless has to be compatible with the “greater-than” of the reals. That’s called a “generalization”.
The way mathematicians came up with a more general “greater-than” – that notably might not align with your intuitive real-based “greater-than” – is to have two sets of elements A and B and if you can make a pair of each element in A with one element of B and some As are left over, the number of elements in A is defined to be greater.
For example if there are some men and some women in an old-fashioned dance event where only mixed-gender pairs are allowed and they all pair up, but some men are left over – then we know the number of men was “greater”.
This allows us to compare two sets that are both infinite and it still can turn out that one set is larger. Not in the lay-stick-besides-each-other-sense, but in the pair-up-sense.
When you have the set of all natural numbers (1, 2, 3, …) and the set of all integers (…, -2, -1, 0, 1, 2, …) then they can be paired up, so both infinities are equal. One way to pair them up would be: 1&1, 2&0, 3&2, 4&-1, 5&3, 6&-2 and so on.
When you have the set of all integers and the set of all rationals (= fractions), then they can be paired up as well.
But when you have the set of all rationals and the set of all reals, then no matter how you pair them up, some reals will still be left over. Both numbers are infinite, but one is larger than the other.
Think of “infinite” as a property like “even” in this case and not of an identity. *Of course* two numbers have to be equal if they are identical.
Imagine you win a contest and get a literal truckload of Cheerios. The catch is, the Cheerios are all poured into the back of a truck, without individual box packaging. The point is, the number of individual Cheerios in the truck is so huge you can’t count it all.
Imagine you can count up to a million Cheerios by hand. Beyond a million, you just can’t keep track of the number of Cheerios for some reason – you don’t have enough time, you didn’t have enough space in your house, or you don’t know any numbers beyond a million, or no human can possibly count to a larger number than that. The actual reason doesn’t matter, but whatever it is, you can’t count more than a million Cheerios by hand. In fact, the actual limit doesn’t matter either. It can be a million, billion, or quintillion. Let’s call it a flugelbloop. No human can, practically speaking, count more than a flugelbloop of items.
So if someone asks you how many individual Cheerios you have, the best you can say is “more than a flugelbloop”.
Your friend entered the same contest, but won two truckloads of Cheerios. Your friend also has the same limit on the number of Cheerios they can count by hand.
How many Cheerios does your friend have?
Again, the best you can say is “more than a flugelbloop”.
Does your friend have more Cheerios than you? Yes, almost twice as many. Because you can literally take one Cheerio from each of your friend’s two trucks, for each Cheerio from your truck.
But both of you have “more than a flugelbloop” of Cheerios.
How is this possible? Because “more than a flugelbloop” is not a number. It is a concept that means “too many to count”.
In the same way, infinity isn’t a number. It’s a concept that means “too many to count” in a particular mathematical Sense.
You can make a set of things with infinitely many things, in many different ways. In the same way that you took two Cheerios from your friend’s trucks for every one of yours, you can actually prove that some of these sets have more things than other sets, though they all contain infinitely many things.
So many wrong answers here…
The simple truth is, two sets are the same size if we can have a one-to-one correspondence between their elements.
All “fours” are the same size because you can do that. Four cats vs four tennis balls are the same size because you can pair them up.
You can do that with infinite sets too. There is the same amount of number 1, 2, 3, … and multiples of five because you can pair them up like 1 with 5, 2 with 10, 3 with 15 etc.
But consider the set of all numbers between 0 and 2, including the irrationals. Turns out you cannot produce a correspondence between those numbers and 1, 2, 3, … Try as you will, there will always be numbers not appearing in your correspondence. This can be proven, and the proof us fairly simple.
In other words those two sets though both infinite don’t have the same size. And moreover there’s not only two different infinite sizes. For every infinite size you can find one even larger.
Let’s say we have two bags of marbles, and we want to know if they both have the same number of marbles, BUT we do not know how to count. How can we check if they have the same amount without counting? We can take one marble out from bag A, one marble out from bag B, pair them up. Keep doing this. If every marble in bag A can be paired with a marble in bag B, with no leftovers, then we know both bags had the same number of marbles.
That’s how we can compare sizes of things without counting them, and how we can compare sizes of things that are “infinitely” large.
In math, we call those bags “sets”. Let’s start with two bags of infinite size that ARE the same size- consider a bag of all the positive whole numbers (1,2,3,..) and a bag of all the positive EVEN whole numbers (2,4,6,…). Since these “bags” contain an infinite number of objects, we cannot “count” how many there are in each to compare the size. So, we have to make pairs, like we did with the marbles. In this case, for every number in bag A, we can pair it with a number in bag B that is twice its value. 1 gets paired with 2, 2 gets paired with 4, 3 gets paired with 6, etc. You can see that for EVERY number in bag A, we can pair it with a number in bag B. So the “size”of all positive integers actually = the “size” of all positive EVEN integers.
Now, there are some “bags” of numbers where it is impossible to make these pairings. No matter how you can pair up numbers, there will always be some numbers leftover that can’t be paired up, meaning that one infinity contains more objects than another infinity, making it “larger”.
This is where it can get hard to explain an example, but we’ll give it a try anyways. Let’s look at these two bags of numbers: bag A will be all positive integers again (1, 2, 3,…) and bag B will be all the possible numbers between 0 and 1 (so for example 0.5, 0.51, 0.501, 0.837362773833333, 0.333, 0.33333333, 0.33333333333333333 repeating, you get the idea- basically all decimal numbers between 0 and 1).
For the sake of argument, let’s say we have come up with some pairing of the numbers in the two bags, and I will write out the first few pairings:
1 with 0.53827263727173000000010000…
2 with 0.8173637363839000000040000…
3 with 0.8387262222233474633000000…
4 with 0.3333333333333333333333333…
Imagine this list being infinitely long, exhausting all the marbles in bag A. However, I can ALWAYS come up with a number from bag B (the decimal number less than 1) that is guaranteed to NOT be on this infinitely long list. I will call that my magic number, and I will construct that number following this rule: I will start at row 1 and look at the digit in position 1 after the decimal point, (so 5 in this case) and for ease of illustration, +1 to that digit and append it to my magic number. At row 2, I will look at the digit in position 2 (1 in this case) and +1 to get 2. Continue down the list, and my magic number will start being 0.6294……
Remember that this list is infinitely long, so if we kept doing this, we would get some decimal number. HOWEVER, and this is the cool part, that magic number is GUARANTEED not to be in the original list. How? Well, let’s go down the list and compare it to all the numbers. Is it the same number as the decimal in row 1? Well it can’t be, since I altered the first digit. Is it the same number as the decimal number in row 2? Well it can’t be, since I altered the second digit. Is it the same number as……? You may start to see the point. So what we have shown is that is impossible to pair up positive integers with decimal numbers between 0 and 1, because no matter how you try to list all the decimals out you can always find a NEW decimal that was not on your original list. This means that the size of the bag containing all the decimals between 0 and 1 must be bigger than the size of the bag containing the positive integers, even though they are both infinitely large.
This is just one example of two different sizes of infinity, but there are many other cool examples that illustrate this. These concepts of infinite have always been one of my favorite things in math 🙂
This one helps me understand better: there is an infinite amount of numbers between 1 and 2. There is an infinite amount of numbers between 1 and 3 as well but this infinity has to be twice the earlier infinity.
The main idea behind bigger infinity is that, with such big quantities, we rely on comparison and not actual “counting”.
If I can show that no matter what I do for every element of X there are 10 of y, y is definitely larger, right?
Now, go to Geogebra and plot the function y = X and y = x^2. Remembering that if a function “is over” the other then it’s bigger, x^2 is basically always over y=x and the more you go towards right and more the difference become larger. How larger? We can know it with a ratio!
Lim X -> infinity of X^2 / X = X and indeed, at infinity y=x^2 is infinitely larger that y=x.
This is not exactly what we means with bigger infinity but I hope I made sense in what comparing infinities means.
Infinity isn’t a single concept, but rather a way to describe things that go on forever. There can be different “sizes” of infinity, even though our brains struggle to grasp this idea!
A list of all numbers ending in 4 is infinitely long. 4, 14, 24, 34, 44… A single value for every set of 10 numbers, going on forever.
A list of all numbers ending in 3 should be the same right? 3, 13, 23, 33, 43… A single value for every set of 10.
A list of all numbers ending in 3 OR 6 is infinite as well. 3, 6, 13, 16, 23, 26, 33, 36, 43, 46… But it has two values for every set of 10.
Both lists are infinite, as numbers don’t end at any point. There’s no “biggest number”. But if you imagine an end or compare a set range, by necessity the list of 3&6 should have 2x as many values as the list of 4.
The numbers between 0 and 1 are infinite if you include decimals.
The numbers between 0 and infinite are also infinite however as they also include all the decimals between each number this Infinate is larger than the former.
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