Can someone explain the concept of “larger infinities” in a way that’s easy to understand?
#infinity #mathematics #conceptualization
Seeking clarity on why infinities can be compared in size – help needed!
#math #infinity #conceptualization
Why do some infinities seem larger than others? Exploring a confusing concept.
#mathematics #infinity #confusion
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
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.
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.
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
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
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
The different sizes come from countability and uncountability. The counting numbers are the natural numbers 0,1,2,… We can prove that all integers, all coordinates, all rational numbers and all rational complex numbers are the same size as the naturals. We call that countable.
However if we consider infinite decimal expansions, we get an issue where with the Cantor diagonalization argument you can’t list them all. So you can’t match a natural number with them, they’re not the same size. There are more size differences between uncountable sets too because you cannot create an injective or surjective function between the power set of a set and the set.
You are getting such bad answers here that I feel compelled to write something.
Lets imagine you have a big pile of marbles and so does another guy. You want to see who has more marbles. Obviously you can just count yours, and he counts his, and you compare the results. But here is the catch: The other guy only speaks only French (and you don’t). So if you try this then neither of you will understand what the number was that the other person reached.
Here is a better idea. Instead of counting marbles, you iteratively roll a marble out of your pile. He does the same. You continue until one of you runs out of marbles. Whoever has marbles left at the end is the one who has more marbles.
The second method of comparing sizes still makes sense with infinite sets so this is how mathematicians talk about the “size” of a set (we use the word cardinality). Of course you might simply guess that when comparing infinite sets using the second method, both piles will always run out of marbles at the same time. It turns out that this isn’t the case. The most famous example is the set of reals and the set of naturals. In this example, the naturals run out of marbles before the reals. Hence, we say that the reals are a “larger” infinite set than the naturals.
You’re reading too much into it. Infinities don’t physically exist, they are just a mathematical toy. It’s just saying “This is the way we generate numbers and in this way we can generate an unlimited amount of numbers”.
“Larger” in this case is also a mathematical trick. It’s not your normal everyday “larger”. We can’t compare infinities like we can compare numbers. So we say okay if we can somehow define a “bijection” between two infinite sets then we call them “equal”. Bijection means each element of these two sets has exactly one pair in the other set. If we can’t do this then the one that always has leftover elements is “larger”.
So you follow this train of thought and the conclusion is under this specific definition of “larger”, some infinities are larger. It’s not something you can visualize.
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.
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 🙂
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.
Don’t think of it as a number. Think of it as something that keeps going. Say you have two cars. Both cars will drive forever. One goes 10 mph and the other goes 50 mph. When go to to check the distance after an hour who drove farther? 10 hours? 1000 hours? Infinite only means they don’t stop. The faster car will always be ahead.
Hi OP, you are getting a lot of wrong answers. Answers that are saying things like “all the even numbers plus all the odd numbers is bigger” are wrong. And “all the fractions are infinite and all the numbers are infinite, so together they are more infinite” are also wrong. The fact that you are struggling with this concept shows that you have a better grasp on the idea than those people.
I’m not confident enough in my own ability to explain it, I’m not a mathematician. With infinities, the only way to prove there are bigger ones is to say “well, if there *weren’t* such things as a bigger infinity, *this* wouldn’t work. But it does, so… Alright.”
It requires conceptualizing infinity in a non-intuitive way. Sort of like conceptualizing “imaginary numbers” (e.g. square root of negative one) requires a different way of conceptualizing numbers. Sometimes in these higher math concepts your intuition needs to be chucked out the window.
Some infinite series we can make a list of. For example 1,2,3,4,5… we never actually complete the list because it’s infinite but if we had an infinitely long list it would contain every number in the series. Some series are too large. If we look at decimals between 0 and 1 we can prove that an infinitely long lost would be incomplete
First, start by making a list of all positive integers: 1, 2, 3, 4…. Yep, all infinitely many. And then assign to each one of those infinitely many numbers one, unique, real value between 0 and 1. So, maybe you assign 0.98989898… to the number 1, and sqrt(2)/2 to 2, and…etc. For the sake of saving time, we’ll pretend you’re already finished, I’m afraid I don’t quite have infinite time to wait.
If there is only one “size” of infinity, then (by definition) your list must contain all real numbers between 0 and 1. That’s why we built it, after all: to make a perfect 1:1 match. Every integer gets exactly one *unique* real number, so if the two sets are the same size, that means every real number must get assigned.
But: what happens if we prove that there’s a real number that *must* exist, but isn’t on your list? That would mean we would have found a number you *couldn’t* index, even with “infinitely” many places to put it. It would mean that there is a new, *bigger* infinity than the infinity we were talking about before.
But that’s crazy, you might say. You made an *infinite* list! How could it possibly be missing anything? There is One Weird Trick (Set Theorists Hare Him!) brought to us by Georg Cantor. Let’s make a number, which we’ll call A. A has digits; these can be seen as a1/10 + a2/100 + a3/1000 + … + a_n/10^n with n running off to infinity. That’s how decimals work, nothing new there.
Here’s the trick. Make a1 something that is *different* from the first digit of the first number in your list. Make a2 different from the *second* digit of the *second* number on your list. Make a3…etc. Repeat this process for every one of the infinitely many numbers in your list.
We have just constructed a number which cannot be on the list: it is, by construction, different from *every other number* in at least one decimal place. This contradicts the assumption that the two sets can be matched up perfectly. Even if you try, you can always generate a new number that isn’t on the list.
This is why we talk about “countable” vs “uncountable” infinity. As Dr. James Grime of Numberphile says, it might be better to think of it as “listable” vs “non-listable” infinity. You can make a “list” of the positive whole numbers. You can’t make a list of all the real numbers; it isn’t possible, doing so leads to contradictions.
Both things are “infinite”, in the sense that they are bigger than any number you could reach by counting up one at a time. But they aren’t *the same size,* because if they were the same size, you could line them up so their parts matched, but they don’t match and cannot ever match. There are infinitely more infinities inside just the range from 0 to 1.
>It’s like saying there are 4s greater than 4 which I don’t know what that means.
Well, that _wouldn’t_ make sense, because 4 is a value, it’s a precise, defined thing. Infinity is not a value, it’s a concept, and in specific terms for the idea we’re discussing here, it’s a set.
This isn’t exactly the most accurate portrayal, but it’s the best ELI5 example I can come up with:
Let’s say you have a set containing all numbers greater than 5. That’s an infinite set, there is no largest number greater than 5. Now let’s say you have another set that’s all positive numbers. Also infinite, but it contains 5 more values than the first set, 1, 2, 3, 4 and 5.
The main infinity we think about (1, 2, 3,…inf) doesn’t end.
But the bigger infinity doesn’t even *begin*.
What number comes after 1? 1.1? 1.01? 1.001? On and on and on, keep going, you’ll never find the next number in the series of numbers between 1 and 2.
So that infinity is so big that you can’t even find where it starts, much less where it ends.
What we refer to as “infinity” is usually a concept – the biggest number possible. There’s no such thing as this number, because any number you pick, I can add 1 to and make a bigger number. So infinity is an idea, not a number.
As to “other” infinities, imagine the number line and focus on the area between 2 and 3. You can put numbers in between these two (2.1, 2.2, 2.3 etc. or as fractions 2 1/2, 2 1/3, 2 1/4…) and there are an infinite number of numbers you can place between any two rational numbers (anything you can represent as one number divided by another), but in our example, all of these infinite number of numbers are bigger than 2 and smaller than 3.
I would argue that in fact the idea that there is “one” infinite is totally counter-intuitive and just something we (think we) learn in school. Doesn’t it make intuitive sense that if you go as far out as you want, you can always go further or take a step back, and so you’ll be a slightly different amount of infinite away from your starting point?
(Mathematicians will probably rightly crucify me for this. But OP is five! What are you going to do?)
Actual mathematician here: half of the responses are completely wrong. While the current top-rated one is perfectly fine, I thus also want to add a proper response:
When you say “infinity” you probably actually talk about the _size_ of things, not infinity as a “number”. We say that two collections (sets) A, B of objects have the same size if we can pair them up: each member of A gets one of B and vice versa.
All groups of 4 objects have the same size. The list 1, 2, 3, 4, … of natural numbers is however infinite and it turns out that a lot of sets have this size. For example the even numbers 2, 4, 6, 8, … can be paired with it:
– 1 <-> 2
– 2 <-> 4
– 3 <-> 6
– 4 <-> 8
– …
A maybe even simpler way to imagine this size, the _countable sets_, is as those of which we can have a neat infinite list. Maybe less obvious is that even all positive rationals, the fractions, can be listed as well. To achieve this you have to sort not just by their actual size as numbers; instead you check which of numerator and denominator is larger and sort by that:
– 1/1, (fractions with a 1 in them and no entry bigger than that)
– 1/2, 2/2 2/1, (fractions with a 2 in them and no entry bigger than that)
– 1/3, 2/3, 3/3, 3/1, 3/2, (fractions with a 3 in them and no entry bigger than that)
– 1/4, 2/4, 3/4, 4/4, 4/3, 4/2, 4/1, (fractions with a 4 in them and no entry bigger than that)
– 1/5, 2/5, 3/5, 4/5, 5/5, 5/4, 5/3, 5/2, 5/1, (…)
– …
By putting all into a single line we get a list: 1/1, 1/2, 2/2 2/1, 1/3, 2/3, 3/3, 3/1, 3/2, 1/4, 2/4, 3/4, 4/4, 4/3, 4/2, 4/1, … which proves that there really are not more fractions than natural numbers!
But are all things “list-able”, or as mathematicians call it, _countable_? It turns out that the answer is NO. The numbers in the interval [0..1] for example can be shown to be so large as to be _uncountable_: there is absolutely no way to put them into a list!
Lets see why:
Assume that some super-intelligent alien arrives and gives us what is supposedly a full list of all numbers between 0 and 1:
– 0.**3**236819479348…
– 0.9**2**83988449999…
– 0.11**1**1111111111…
– 0.879**9**547771234…
– 0.0367**2**36472838…
– …
Lets prove they are a dirty liar! I’ve marked some decimal digits in bold: the first of the first number, the second of the second number, and so on. They together spell a number, **0.32192…** which might be somewhere in that list. But now change this number a bit into **0.43203…** where we changed each digit into the next larger one (and 9 into 0). Note, and this is the most important thing about our fancy new number, **its n-th digit is different from the n-th digit of the n-th number on the supposed list!**
Therefore this fancy number cannot actually be on the list! Say it is at the 1,000,000-th place. But the 1,000,000-th digits of our fancy number and the 1,000,0000-th on their list do not match up. It cannot be there, nor can it be anywhere else. We found a smoking gun once and for all proving them to be wrong!
In short, there are sets with sizes beyond the countable range. And one can even show that there is an infinity of infinite sizes!
As a side-note: there are also completely different ways to have infinities as actual _numbers_. They then do **not** represent sizes of things anymore, they are just that: numbers, things we can calculate with, doing their own thing. Even in the finite realm not every number is the size of something (or show me something of size -0.12345…. !).
Then with ∞ as an actual number, your question becomes surprisingly boring: obviously ∞+1 is larger. That’s it. It isn’t very enlightening, just true.
> Edit: the comments are someone giving an explanation and someone replying it’s wrong haha. So not sure what to think.
Yeah a _lot_ of people in this topic responded while having absolutely no clue what they are talking about. No idea why they felt compelled to.
Infinity isn’t a number, so saying that some infinities are bigger than others isn’t like saying some 4s are bigger than other 4s. We’re talking about something else entirely.
You need to fully understand the concepts of cardinality, injective, surjective, and bijective sets first. Do you understand those yet? Sets have the same cardinality if there is a bijection between them. There are proofs that some infinite sets do not have a bijection between them. Therefore they are not the same size (do not have the same cardinality).
Some infinities have a clear starting point, and you could reasonably count up pretty far if you had the time. While with others you can’t even get to number 2 because you’re stuck counting *the zeroes* on your way to your very first number.
>It’s like saying there are 4s greater than 4
Don’t think of infinity as a number or quantity. When you do that, you already *assume* that there aren’t degrees of infinity.
Infinity is more like a description or a property or an adjective: things that are *infinite* have no *end* (they are ‘*not finite*’).
So, we look at different things that *don’t end* and instead of assuming that they are the same size, we investigate with an open mind to see if they are the same size.
—
In Mathematics, we say that two collections of things (‘sets’) have the same size (‘cardinality’) if we are able to pair up the things in those collects (the ‘elements’ of the ‘set’) in a one-to-one way.
For finite sets this is easy: you can just compare the total number of items in each set and see if they are equal, and if so, then you could pair them off easily.
For infinite sets, it is difficult, because there *isn’t* a number that describes how many there are, and that’s why we try using this ‘pair off’ concept.
For instance, there are infinite counting or ‘natural’ numbers (1,2,3,4,5 etc). And there are infinite even numbers (0,2,4,6,8 etc).
There are many ways to imagine trying to pair them up one-to-one with each other, and if there is at least 1 way to pair them up this way, then we say they are the same size.
For the natural numbers numbers and the even numbers, it is easy to do. We can simply list the even numbers starting at 0, and count up by 2 each time, like this:
1. 0
2. 2
3. 4
4. 6
5. 8
6. 10 …
[etc, where for each natural number *n*, then we’d say that the *n*th even number in the list is (n-1)*2.]
Indeed, any infinite set where there is some way to *list* out each element numerically will be the same size as the natural numbers, because for each natural number there is a way to pair it up with the spot on this list.
—
Well, now the question is if there are any infinite sets where we cannot list every item? Is there some set where no matter how cleverly we construct our list, some numbers are impossible to include?
It turns out that there is. An important one is the ‘real’ numbers (all the numbers including irrational numbers like sqrt(2) and pi and e and many more) can’t all be listed.
You might think “well, just imagine a list with all of them”, and that sounds like it could work, but unfortuantely it doesn’t. No matter how clever a system you use to construct this list, you’ll always miss some off your list, even though your list goes infinitely long. [This is proven by a argument known as “Cantor’s Diagonalisation”, which is a bit hard to explain quickly, so for now you need to trust me, but you could research it if you like. The jist is that Cantor imagines you have an infinite list of infinite decimal numbers, and uses it to generate a number that is is missing from your list, and he can *always* repeat this no matter what list you present to him, and so your list *cannot* be completed.]
You might then think “I’ll just add the missing numbers to the start of my list” or “let’s alternate between my old list, and one of the missing numbers, to build a new list”, and that sounds like it should work, but this too fails. The former works if you have a finite amount of numbers to add, and the latter works if you have two lists you want to combine. However, the vast quantity of missing numbers are such that *any* infinite list cannot get them all.
Okay. I am going to take a stab at this.
Imagine all infinite numbers are in the shape of a ball, the diameter is the “size of the infinity”
Let’s look at all the numbers between 0 and 1 you have an infinite number between that but it ends at 1 so your ball is 1 diameter.
Now think that you can go to -1 and 1 this has 0 and 1 AND 0 and -1 which is a “2” diameter. It would be twice as big as a ball but it still has an infinite amount of numbers in it.
You can then extend the ball size to ALL positive numbers. This is an infinite sized ball and very big, but as we saw we can make a bigger infinite ball by having one with All positive and ALL negative numbers. It is still infinite but it will be TWICE as big!
Hope that helps.
Math major here!
First, don’t think of infinity as a number. Think of it like a briefcase. It sort of holds/subsumes all the other numbers. Obviously, there are different kinds of briefcases that all do unique things for us.
The first briefcase we have is called a “countable” infinity. This is the kind where you can just list everything out in some logical order. Imagine you’re a lawyer preparing for a huge court case. Your briefcase needs to contain all your documents and laptop in a manner so that you can get everything exactly when you need it. That’s countable infinity. It’s literally everything but arranged in such a way that we can make sense of it. An easy example of this, is whole numbers 0, 1, 2, and so on.
Our next briefcase is called “uncountable” infinity. This is the bigger one. We call this kind of”uncountable” because you can’t easily list everything in this briefcase. It is so crammed full of documents that you would never be able to sort everything out and that’s why we say it’s the “bigger” infinity. An easy example is all the numbers between 0 and 1 like 1/2, 0.0000000000000000000000007 and e/pi. This list of numbers is so unimaginably huge that mathematicians don’t even bother really inspecting the items here. They just group it all together under what we call Aleph-1 (that’s our briefcase) and go about their business.
>like saying there are 4s greater than 4
Imagine you have an infinite number of 4s. Now imagine you have an infinite number of 5s. Which is more?
Infinity isn’t a number, or a set amount, there are different levels of ‘infinite’ rather than one consistent ‘infinity’, an infinite value is just endless or uncountable, not all infinite values are the same, and they each can represent different things
Imagine a hotel with an infinite amount of rooms, if each room has 2 beds, then there are also an infinite amount of beds, but the infinite amount of beds is double the infinite amount of rooms
so you have two infinite values with one being bigger than the other
If it makes you feel better, I find it to be a meaningless pedantic topic precisely because Infinity is simply unquantifiable. How you go about measuring an unquantifiable concept has no real bearing on anything. In each of these logical thought problems, Infinity is always the remaining set no matter how much of the set you’ve already counted.
Eli5: Countable infinity is for example a set of natural numbers. If I tell you to count from 1 to infinity using only decimal numbers, you start with one, then go to two, …, a billion and one, …, Infinity-1, infinity. Cool this is infinite large set of numbers, you cannot really get there, but you at least know how you could get there.
Uncountable infinity: Now I tell you to tell me all the real numbers between one and two, well you start at one and then,… what the hell is the next real number after one, it is smaller then 1, smaller then .1 smaller then 0.0000001, so not only is there infinite number of them but it is also basically impossible to count in this direction.
The thought experiment that hints at why uncountable is larger than countable goes like this
Lets try to make a dictionary matching them.
Countable – uncountable
– 1 : 1.000{Infinite number of decimal places}0
– 2 : 1.000{Infinite number of decimal places}1
– 3 : 1.000{Infinite number of decimal places}2
– N : 1.{N-1 number of decimal places}5{infinite nod}5
– Infinity minus 1 : 1.999 {Infinite number of decimal places}9
– Infinity : 2.{Infinite number of decimal places}0
Same size? Well the though experiment now goes. What if I construct a number by taking a digit from each real number corresponding to its index position and increase it by one (or rollover)
Ie:
– 1.1 (because of index 1)
– 1.11 (because of index 2)
– 1.111(because of index 3)
– 1.{N-1 number of decimal places}6 … becase of index N
– 1.111…6…0 (because of infinity-1)
Well if you think about it, this number is not in our dictionary, it cannot be, because at least one number (the one on diagonal) differs for each corresponding row. Which means that our dictionary is incomplete, you cannot match all elements of uncountable infinity to a a member of countable, there is just more of them.
Think of infinities like…. shooting something into space. Something flying through space goes on forever, right? Well what if one thing is shot into space faster than another thing? That would be considered a ‘larger infinity’. They both go on forever but one is clearly further than the other.
Another cantor to the questy is some infinities are countable and some just not countable at all.
I remember reading something like infinity is not a destination, it is a journey. Since infinty is not a discreet value like 4, it would not be wise to compare it as such.
Imagine you have two spaceships with the ability to travel forever. Both start in the same position, however one ship travels at a constant rate of say 20km/h, and the other travels at a constant rate of 10km/h. Now since they can travel forever, the distance traveled for both is “infinity” however at any time t, one ship will always be farther away from the other. So one infinity is bigger than the other.
Another thing to consider is 4/4 is 1, however let g(x) and f(x) be functions that go to infinity. G(x)/f(x) may not be equal to 1.
Infinity is just a concept that describes certain types of number sets.
Countable infinity is all the numbers
Uncountable infinity, or infinity squared as I understand it, is all of the numbers between the numbers. Like 1,1.00000000001, etc.
Uncountable infinity HAS to be larger than countable infinity because it includes all of the numbers in countable infinity and then has an infinity in between each of those numbers. There’s simply more numbers.
Infinity as a concept is just either the sum of the entire data set or the number of numbers in that data set.
Some infinities are like a runner that never stops. The number keeps getting bigger.
Some infinities are like a camera zooming out, so you can see more of both sides.
Some infinities are like a camera zooming in, closer and closer, yet getting more and more detailed.
A simple but wrong answer is that between 1 and 2 there are infinite 1.xxx numbers.
So before the infinite set of real numbers has even counted to 2, we’ve an infinite set of decimals between 1 and 2.
I think that’s probably technically wrong but it’s a useful way of thinking about it.
I like to think of “infinity” as of “a lot” instead of an actual number
I have an infinite amount of balls = I have *a lot* of balls. How many exactly, I don’t know for sure, but I do have a lot.
Now if I have a lot of balls and my friend also has a lot of balls, it could be that he has way more than I do, of perhaps way less, or the exact same amount. There’s a degree of imprecision that cannot be counted.
The same principle would apply to a take away. I have a lot of balls, but you take away a lot of them. Do I still have a log left (meaning my initial infinite is much larger than the one you took away), or do I have very little?
I hope this helps!
Because mathematical “infinity” is different from linguistic/conceptual/philosophical “infinity”
Wait till you find out that, thanks to infinity, 0.99999… (repeating infinitely) is actually the same number as 1. Not “almost but not quite” or “close enough to consider the same” — they are quite literally the exact same number.
It might make more sense to say there are different kinds of infinities than to say there are different sizes
String and apples. I can count apples 1, 2, 3….and as mental exercise keep going forever. I can get a length of string and again as an mental exercise extend that string forever. Those are two different kinds of infinity.
This isn’t a proof but I can come up with a length of string I can’t match to a collection of apples or a ratio of two collections of apples: I can arbitrarily define a length of string as 1 unit. I can use that length of string as sides of a square, and then the length of the string along the diagonal is square root of 2 via the Pythagorean theorem. But we know that the square root of a 2 can’t be a rational number (in our analogy a ratio of 2 collections of apples. So we have a length of string that can’t be described in terms of apple collections.
There’s more to it of course but I hope this gives you an intuition that there’s more than one kind of infinity.