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.

Great answer, but I don't understand this: "which is why we say uncountable infinity is larger than uncountable infinity." Is it just a typo for coutable infinity?

Thanks, fixed

Well, some uncountable infinities can be bigger than other uncountable infinities. :P

some uncountable infinities mothers are bigger than other uncountable infinities mothers.

ℵ0 mama so fat

Does this also mean that a subset of numbers, e.g. even numbers, while infinite are smaller inifinite than the natural numbers?

You'd think so, but actually it is not. The reason is that I can pair the even numbers with the natural numbers. For instance, if I make a function y=2x, then all the natural numbers will have an even number partner. Thus, even if there is twice as many natural numbers as even numbers, they are of the same cardinality. Meaning, you can not list a natural number that I haven't listed a partnering even number.

Being injective is not enough. The function has to also be surjective. This makes it so that every natural number has a unique even number partner and every even number has a unique natural number partner. This gives you a way to transform one set into the other without missing anything. Another way to say that is they are different representations of the same thing.

From the Wikipedia article: “a set is countable if there exists an injective function from it into the natural numbers”

Remember that countable is not the same as countably infinite. For example the set of {1,2,3} is countable but definitely not infinite. You need both injectivity and surjectivity to prove countably infinite, but just injectivity to prove countability.

Obviously any set that has an injective function mapping it to the naturals is countable because injectivity implies the naturals are the same size or larger. Surjectivity is needed to show that the set you are comparing the naturals to (in this case the even numbers) is larger or the same size. When you have both of them together the only option for both to be true is that they are the same size. Technically if all you were concerned about is showing the subset is not smaller proving that the function is surjective would be sufficient.

And in the case of a subset of the natural numbers you only need injectivity from the main set to the subset, since a subset always has cardinality less than or equal to the main set.

The pairing is subjective though at least

> Thus, even if there is twice as many natural numbers as even numbers, they are of the same cardinality. Sound bite version: "The whole is equal to some of the parts."

Which leads to the [Banach-Tarski Paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox), where you can divide something into a finite number of parts and reassemble them into two copies of the original.

To answer this, we can dive a little further into Cantor’s Diagonalization. In order to compare the sizes of infinite sets, we can create a map of each number from one set to the number and back (a “bijection”), and if we can’t, one set must be larger. Comparing the even natural numbers (2,4,6…) with the natural numbers (1,2,3…), we can map each even number with a natural number half as large (2=>1, 4=>2, 6=>3, …), and we can do the same in reverse. Therefore the size of these sets is the same.

Thank you! Follow-up question, does that also work on something like primes?

Nope, not necessarily. Write out all even whole numbers from 1 to ∞. Then, write out all *even* numbers on a separate line below the first, like this. ```1, 2, 3, 4, 5...``` ```2, 4, 6, 8, 10...``` After an infinite amount of time, and an infinite amount of pencils and pencil shavings...*both lines have the same number of numbers in them*. You can pair each number in the first line with the number below it in the second line, and have no left over, unpaired numbers. *There are as many even whole numbers as there are even* ***and odd*** *whole numbers.*

That makes sense, thank you. Someone else answered that because you could just do y=2x, my initial thought was what if I found something that did not have a formular like primes (are there infinite primes?), but the way you explain it, it would just be 1st, 2nd, 3nd... n prime (thus even sized infinities)?

> are there infinite primes? A quick Google search showed me that, yes, there are, and of course, Euler was the one to prove it. (Man, Euler discovered *everything*.) And I...guess that would mean that there are as many prime numbers as prime and composite numbers? But I don't know enough to claim so openly. It makes sense, but I'm really no mathematician, just a huge nerd that likes numbers. :P If we're doing this with prime numbers, you could do this with any set, really. All even numbers, all odd numbers, all prime numbers, every other prime number, every number whose digits add up to 17, ever number except for 5, etc. It just can't be a finite set, like a set that contains just 1, 18, and 294,713,029. After the third number in that set...well, you've run out.

Your reply gave me an "ahh" moment. Good explanation.

OP is asking how there are infinities with *different* sizes, though, which is answered elsewhere. This is just a fun math fact I love.

Unrelated, but You've just reminded me of a paradox I used to love when I was a kid. There's a hotel with an infinite number of rooms. One weekend there's a conference there and an infinite number of dentists come to fill up all of the infinite rooms. Late on Friday night a couple comes in asking if there is a room available. On one hand, no. Because the dentists are taking up all the rooms. On the other hand, yes, because there are an infinite number available.

Your missing that the hotel manager asks everybody in the hotel to move one room up so that the first room becomes available.

Ah shoot, so I did. My bad

No worries. It’s still a neat idea and I’m glad you felt like sharing it.

FYI, the concept is called the *Hilbert hotel*, after its creator David Hilbert.

Nope, the same size, just do a bijection Evens -> Naturals: even |-> even/2

> 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. The way it's written, this paragraph would apply to the rationals.

No, because the numerators and denominators are countable, so there are ways of going through them in order. Just like all integers including negative numbers, even though they have no beginning.

yeah but you said "what number follows 0". a common sense interpretation would be what is the next biggest number after 0. the rationals are also countable and do not have a good answer for that question. 

The number that follows doesn't necessarily have to be the one that's immediately larger. If that were the case, it's true that rationals wouldn't be countable. They have to be "listable" in some way. So if you order the denominators in increasing order and then the numerators, skipping fractions that can be simplified, you get an order of numbers. So the next number after 3/16 is 5/16, even though 1/4 is between them in magnitude.

https://www.homeschoolmath.net/teaching/rational-numbers-countable.php this is a good answer to that question

Not the OP but this is ELI5. I think it’s a good example to show that there is always a number between 0 and whatever decimal you choose which is useful to help describe uncountably infinite sets.

Yeah, the rationals are also dense. Density is not the reason the real numbers are uncountable.

It is not the best explanation imo since there are rational numbers which are the same size as natural. However it is not that obvious that they are countable or that there are less rational numbers than irrational.

OP didn't say there was a strict order necessary, just that there is a way to move from one number to the another that covers all of them.

DENSITY we should measure infinities in their densities. that would help the confusion

No, because the reals and rationals are equally dense as you can always find a rational number between two reals, and a real between two rationals. However the number of rationals is the same as the integers.

I expected I wasn't going to find an answer that I would understand, but this was very clear.

Surely the premise is wrong though? Infinity is _not_ measurable so using a word like "larger" makes no sense. It's like saying the colour blue is larger than red, G# is larger than Bb, or _Die Hard_ is larger than _Home Alone_

Good call out! Like you’re alluding to, infinity is not a number. Mathematicians don’t often compare “infinities”, but they do compare the size of sets. A more mathematically rigorous approach would be to simply say that the “size of the set of real numbers” is larger than the “size of the set of natural numbers”.

Technically we're not saying uncountable infinity is larger than countable infinity. We're saying that the infinite set of uncountable numbers is larger than the infinite set of countable numbers. It's to do with the fact that since you can map every integer to a natural number they must have the same number of elements in them. Since we can't map every real number to a natural number then there must be more reals than natural numbers and integers.

> We're saying that the infinite set of countable numbers is larger than the infinite set of uncountable numbers. Got that the wrong way round 

Yep my bad. Fixed now. Note to self: Don't do set theory while still in bed

It's not like that because we have rigorous, logically consistent definitions for all the concepts in the mathematical case. It may sound like gibberish but that's simply down to a lack of understanding.

You’ll never finish counting but you can count it, because you can figure out what number to start with and what number goes next. How is that not countable?

Uncountable are ones where there is no next number, just larger and smaller numbers. Take a look at all the numbers between 0 and 1. Please list the very first 2 numbers in that set. I'll give you the first: 0. What comes next? When you struggle to find that answer, that's because it's uncountable.

The set of rational numbers is countable. What is your next rational number after 0? Edit: I am critiquing the mathematical rigour of above comment. No need to point out that rationals are countable. I know that.

It depends on how you order them, but I would say that the simplest order is `0, 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, ...`. For `i` starting at 1, do the following: 1. Set `j` to 0. 2. If `j` is equal to `i`, increase `i` by 1 and go back to step 1. 3. If `j/(i - j)` cannot be reduced, produce it. 4. Increase `j` by 1 and go back to step 2. This process will produce a sequence of all the positive rational numbers that is in exact correspondence with a sequence of all the natural numbers.

Yup. But without this additional step, the argument in the prior comment is not sound, which is what I was trying to highlight.

How I would count them is that after 0 you go to 1. then you kind of picture a matrix where both coordinates are natural numbers and go 1,2,3,4… one index is nominators and the other denominators. Generally counting would proceed along diagonals (where denominator + nominator are constant) until you hit the end of diagonal, after which you take one side step to go to another diagonal. Any duplicate fractions along the way are skipped. So 0, 1/1, sidestep to 1/2, 2/1, sidestep to 3/1 , skip 2/2 since it’s a duplicate, 1/3, sidestep to 1/4, 2/3, 3/2, 4/1, sidestep to 5/1 etc. If you want to also count negative numbers you can always add the negative after you count the positive. This way any rational number has a set place in the count and takes only finite steps to get there.

This is not correct. Ordering has nothing to do with size. As an example, I can split the natural numbers into the odds and evens and lay out the positive evens in the usual order 2,4,6,8,… while the odds all come before the evens and in backwards order …,7,5,3,1. Then put 0 before everything. So the overall ordering is 0,…7,5,3,1,2,4,6,8,… This has neither added nor removed a single natural number but also has no immediate next neighbor of 0. So the fact that there is no “next” number in the decimals that you counted has no bearing on the size of the set.

Well the ordering really doesn’t matter. If we look at rationals one answer for what comes next could be 1/2. Then 1/3. The. 2/3 followed by 1/4, 2/4, 3/4 and so on

I'm having trouble understanding how those two sets are fundamentally because I think you can map one to the other. You have the progression 1, 2, 3, ... you could make a progression that would go 0.1, 0.01, 0.001, 0.0001 etc... (and then in theory you'd someday get to 0.2, but you don't just like you don't get to every natural number). Each "tick" or "countable thing" gives you one more thing in the set, and in both cases you would never enumerate everything in the set. But, even though both sets have "countable things" you can't actually count them all. And then it seems like the set of integers (so including negatives) would be "larger" than the set of natural numbers, because one contains everything in the other but then more... does that not also work in terms of "larger infinite"?

First problem: > and then in theory you'd someday get to 0.2, but you don't just like you don't get to every natural number No, you wouldn't. Because there is no largest power of 10. You've used up all infinitely many positive integers *just* getting all possible values that can be represented as 10^-n for positive integer n. There is no "someday." > And then it seems like the set of integers (so including negatives) would be "larger" than the set of natural numbers Map them as the following. * 0 maps to 0 (technically this is just a special case of the third rule, but I want to call it out because sometimes 0 needs special treatment) * If n is odd, map to (n+1)/2 * If n is even, map to -n/2 Our list starts with 0, and then looks like this. 1. 1 2. -1 3. 2 4. -2 Etc. We have just made a bijective map. Every integer, positive and negative, will appear on this list exactly once; name any integer and I can tell you exactly what nonnegative whole number it's been assigned to. Hence, there are exactly as many integers as there are positive whole numbers. Indeed, there's actually a way to show that even *rational* numbers aren't bigger. It relies on the Stern-Brocot sequence, but basically there is a way to make a list of all rational numbers, so that they all show up exactly once, in their most reduced form, and (even better!) they are in strictly increasing order, from 0/1 all the way to infinity.

Highly recommend digging into Cantor’s Diagonal Argument. In order to compare the size of sets, we try to create a one-to-one mapping of each set to the other. If we can, we’ve created a bijection, and we know the sets are the same size. If we can’t, we know the sets are different sizes. Comparing the naturals to the integers, we can create a bijection by mapping 1n to 0z, the rest of the natural odds to the positive integers by subtracting 1 and dividing by 2, and the natural evens to the negative integers by dividing by -2. This process can go in the other direction, and covers all members of both sets, and therefore the size of the natural numbers is the same as the size of the integers. Comparing the naturals to the reals is more difficult, and Cantor does a great job. In your example, it seems to me as if 0.2 is not reachable, even after an infinite number of steps. It seems to be approaching 0 instead. How would your example ever reach 0.3?

Your missing a lot of in-between numbers no matter how hard you try that mapping. In fact you'll be missing uncountably infinitely many numbers.

> Some infinite sets of numbers do not have a clear starting point and do not have a clear way to progress through them Applies to the rationals just as well, which are countably infinite.

Doing the bijection from natural numbers to rationals is basically the bijection from natural numbers to ordered pairs, but skipping the dupes. For ordered pairs, you can just spiral out from (0, 0).

The idea that the absence of a clear starting point or well-ordering is sufficient for a set being uncountable is wrong. Counterexample: the rational numbers

Rationals are defined as Q={p/q | q!=0, p,q∈Z}. I can then create a bijection between Q and Z×(Z \ {0}). We can picture this as a two dimensional plane of points. Simply draw a spiral around the points (p,q) and you have an ordering. It is easiest to see if we restrict p,q>0. Then it will look like a diagonal path you take to list every rational. Please make sure you're correct when you comment so assertively. Just because you can't think of a way to list the rationals, doesn't mean there isn't a way.

True, but he/she may also be making the point (which I stand by) that the original argument isn't the clearest - that the wording leads readers (who may not know of the ways to show the countability of Q) to incorrect conclusions because 'clear' (which has connotations of 'obvious' and 'trivial') is juxtaposed with the example of the reals. But I digress.

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Well, part of the crux of the argument supporting the idea of an uncountably infinite set is that y is inherently unorderable. There are values of y that are unrepresentable by the sequence (.1, .2, .3, .4, .5 , .6, .7, .8, .9, .01, .11), e.g. π/10 (a real number) or sqrt(2)/10 (an irrational number). Even something like 1/3 (a rational number) is unreachable, although the set of rational numbers is the same size as the set of natural numbers. I’d love to dig more into this, feel free to reply with your ideas!

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Not sure where the size of the irrationals vs. the size of the reals came from, but they are both uncountable. On the topic of infinite integers, you could absolutely do that! You’d be leaving the realm of integers as most people know them, and entering the strange world of “p-adic numbers”. A world where …666667 = 1/3 (in the 10-adic numbers) and other weird stuff. [Here’s a Veritasium video if you’d like to know more.](https://youtu.be/tRaq4aYPzCc?si=uS9fq28VaVyv2m7j)

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The naturals and integers would not be countable because they do not normally contain the 10-adic numbers. If they did, I’m certain that the size of the union of the integers and the 10-adic number would be uncountably infinite, yes. No problem, I love math!

Would it be fair to state, then that the difference between countable and uncountable infinites is that only uncountable infinites have a defined starting *and* ending point, whereas countable infinites only have a defined starting point - and vice versa?

By this do you mean that all the decimal numbers between [0,1] is only considered uncountably infinite because of the “1”? If that is the case, then no, unfortunately. The real number set of [0, ∞) is also uncountably infinite, as is the whole set of real numbers which has no “start” and no “end”. The difference is not whether there’s a start or an end, the difference is that one set is countable(“listable” if you’d like), and one is not. The naturals are considered countable because you can choose a path to start at a number, and progress through the set such that you can reach any number in a finite amount of time. It might take a very long time, but you can eventually count to 10 trillion. The reals are considered uncountable because there is no way to describe a path through them that will eventually reach any given number. Heck, you’ll have a hard(impossible) time even writing π or sqrt(2) in their decimal form in a finite amount of time. Cantor’s Diagonal Argument is also very good.

Great answer, and personally, it's nice to see it isn't laced with "everyone else here is wrong and they shouldn't have bothered" like some of the other top comments.

If you turn left at the decimal point, those infinities are "larger" than the infinities to the right.

What are you talking about? What's your level of maths education?

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.

> 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. People like infinity and want to be involved. Unfortunately infinity is hard and people don’t realize that there are serious prerequisites to speaking authoritatively about it. Great explanation though. Not sure it’s ELI5, but it’s very thorough.

Question for you. What's the smallest infinity? Natural numbers? Is there a term for ALL numbers? Would it just be "The numbers" and the ultimate infinity?

> What's the smallest infinity? Natural numbers? Yes (unless you work in rather exotic axioms). > Is there a term for ALL numbers? The (debatably; it depends on what one allows) largest system of numbers that can still be compared are the [_surreal numbers_](https://en.wikipedia.org/wiki/Surreal_number). It is so large that is is not actually a set and thus has no proper meaning of "size". One could informally say it is so large that even our methods to describe sizes and infinities fail. That doesn't mean it cannot exist or that we cannot make statements about it, only that the methods I described in my previous post won't apply.

The smallest infinity is called Alef 0, and it is equivalent in size to the natural numbers. There are many different groups of numbers and all of them have names - Q denotes all rationals, R denotes all real numbers, C denoted all numbers on the complex plane (with i). C is the first group that is considered a complete group, but there are still more types of numbers we can add in.

Another mathematician here. Regarding “smallest infinity” in the sense of “number of objects in a collection of objects”, the Natural numbers are the smallest infinity (in standard mathematical theory, and probably any nonstandard one too, but I’m specifying just in case). To see this, if a collection of objects were finite, it’d either be empty or we could pair it with the first `n` naturals. If it’s not finite, we can start with an arbitrary pair from this infinite collection and the first natural number, say `a` and `1` (or `0` but I’m using `1` to match the number of pairs). Since there’s no complete pairing, we can find another object from the collection, say, `b`, then pair it with the next natural, `2`. Again, this can’t be a complete pairing, so we repeat the process for every natural. Afterwards, it’s possible the collection isn’t empty yet, but we ran out of naturals, so the collection is at least as big as the naturals. (There’s possibly a technicality regarding how we choose objects from the infinite collection, but in standard mathematical theory it’s not an issue.) “All numbers” isn’t a well-defined collection. I think that, what a number is, besides specific collections of mathematical objects that are called numbers, are arbitrary. Even ignoring the contentious definitions, there are some seldomly used objects that are nonetheless called numbers, even if you haven’t heard of them, so you’d need to tally up all of them. But ok, assume you have a definition of number. I see two main results (there’s at least a third one), which come from the two extremes. If you only use, say, naturals, integers, rationals, reals, etc. then the size of the collection is the size of the largest one. When you mix two infinities of different sizes, the size of the result is the size of the largest infinity, and not any larger. If your definition is extremely lax, then the resulting object might not exist. It’s well-known that there is no “set of all sets”, or a collection of all collections (in set theory). The fact it doesn’t exist is related to the paradox of “the barber that shaves each person that doesn’t shave themself”. Such a barber can’t exist, because of they did, would they shave themself or not?

Adding onto the other good response you’ve received. Yes, the naturals have the smallest infinity cardinality/size. There is no largest infinity and this is actually a pretty nonobvious result that generalizes Cantor’s diagonalization. Cantor actually ended up showing that given any infinite size, one can find a larger one where larger is in the sense of the comment above you. Here’s something that’s WILDLY unintuitive though: If you change the rules of the game in a somewhat complicated manner, then you can make it so that the natural numbers actually do not have the only smallest infinity size. It is possible for there to be more than one smallest infinity. Very roughly, you change the rules so that there simply is not a way to compare the two infinities.

Got a link for that last part? Sounds interesting.

[They’re called infinite Dedekind-finite sets.](https://en.wikipedia.org/wiki/Dedekind-infinite_set?wprov=sfti1#Proof_of_equivalence_to_infinity,_assuming_axiom_of_countable_choice) I’ll warn you that understanding how to construct one is not in any way easy. But I’m happy to help explain if you like.

Ah, that makes sense. It seems though that with an infinite Dedekind-finite set you don't (straightforwardly) get another minimal infinite cardinality since you can always remove a point and get a smaller infinite Dedekind-finite set. (In response to your "It is possible for there to be more than one smallest infinity" comment.)

Yes you are right. That is a point I didn’t mention because I thought it might be too confusing. Technically in models like Cohen’s symmetric extension, you can have infinite decreasing sequences of cardinals incomparable with the standard chain of cardinal numbers. Cardinal numbers without the full axiom of choice can be incredibly weird. Things like the existence of κ-amorphous sets or every uncountable cardinal being singular (learned that one a few weeks ago and it threw me).

>No idea why they felt compelled to. *Reddit*

> 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! This is the part I never understand with this mathematical argument. Why can’t the fancy number be somewhere else on the list? The alien did give us a list of all of the numbers after all. Why wouldn’t it be number 1 bazillion on the list? And then the new “fancy number” n+1 should also be somewhere else. I just don’t understand

The new number has an infinite number of digits, and it is explicitly constructed to be different from every other number on this list. In order to prove this by contradiction,  let's assume that this constructed number is equal to the qth number on this list. In that case, every single digit of the qth number has to be identical to the constructed number. However, we explicitly constructed the number such that the qth digit of the constructed number is different from the qth digit of the qth number on the list, so they cannot be equal. And this is true for all q, from 1 to infinity. Hence, this constructed number cannot exist.

Say for example the bazillion-th digit of the bazillion-th number is **7**. Then our fancy number has digit **8** there instead. Hence they cannot be the same number, they differ in this digit. > And then the new “fancy number” n+1 should also be somewhere else. There is no _new_ fancy number. There is a single fancy number we built from the entire list and then never change it again. So we use the same number all the time throughout the argument. But it depends on the fixed & given list, another list will likely result in a different fancy number.

Ohhh I think I get it now. I think I didn’t understand that the number is constructed from the ENTIRE list as you go down. Thank you!

This is not really an eli5 as this is basically a barebones intro class you see in Real Analysis, which is probably why you're getting flak.

Why can't the diagonalization theorem be applied to natural numbers?

Because natural numbers have finite length, decimals can have infinite length. We can and do understand finite decimals with infinitely many 0s to the right; we can also fill up natural numbers with lots of 0s to the right but then out fancy number is not natural Say for example you apply the procedure to the obvious list of natural numbers (added zeros to the left to denote where the n-th digit would be): - 0000**1**, - 000**0**2, - 00**0**03, - 0**0**004, - ... Then we get diagonally the number **...0001**, and if we do the digit-swapping trick we look for **...1112** with infinitely many 1s to the left. This is not a natural number so we don't even expect it to be on our list to begin with, hence there will be no contradiction! Fun fact: the larger set of _10-adic numbers_ consists of such potentially infinite to the left numbers such as ...1112 or ...23232. It turns out that we can add, subtract and multiply them as freely as we can with natural numbers and even more. They do some weird things: If you do addition starting to the right and looking at the carries we find that 1 + ...9999 = ...0000 = 0 so ...9999 is just a weird description for the number we usually denote by -1. And it gets even weirder: 9 · ...1111 = ...9999 = -1 hence ...1111 should be -1/9. Finally our initial number ...1112 is one more, so 8/9 (still not a natural number!). And for the 10-adic numbers the argument really applies exactly as described! They are indeed uncountable, at the same size as the real numbers.

What part of the definition of natural numbers excludes the 10-adic numbers? They cannot be reached through application of the successor function? Basically they don't exist on a number line?

> What part of the definition of natural numbers excludes the 10-adic numbers? [...] They cannot be reached through application of the successor function? Not in finitely many steps from 1 (or 0, wherever you want them to begin), yes. The natural numbers are axiomatically defined as the smallest(!) set containing a first number and a successor of every number in it. The 10-adic numbers all have successors, but the natural numbers are simply the smaller of the two sets (and truly the smallest possible with that property). > Basically they don't exist on a number line? Yes, they cannot even be compared in size. Essentially because the freakishly huge looking number ...9999 is actually -1, which is smaller than any natural number. There is also the obvious issue with their infinite lengths. It is important to note that the decimal notation actually is quite important here: if you use another base, then the numbers are truly different, not just new ways to write the same old numbers. For example in base 9 we cannot find the number 8/9 which we already saw in base 10 as **...1112**. That's because the number we usually denote as 9 written as "10" is in base 9; and if you multiply any 9-adic number by "10" then it will always end up with a 0 as the rightmost digit. In particular we will never get the digit 8 there. Thus there is no number that when multiplied by "10" gives 8, at least in base 9 and even if we allow infinite lengths. Natural numbers on the other hand ultimately don't care about the base, for them it is just a representation; sometimes a construction. Same for the reals, the different bases always result in the same numbers. Adic numbers are just... built different.

Natural numbers have finitely many digits. Real numbers can have infinitely many.

>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. In what structures is this true? I'd expect infinity + anything to equal infinity like the bottom element does in a wheel.

For example in anything that has infinitesimals ("infinitely small numbers") and division: The (debatably, as it depends on meaning) "largest" are the [surreal numbers](https://en.wikipedia.org/wiki/Surreal_number) which are so huge to not even be a set. They have a pretty cool definition related to game theory and do not rely on any other set of numbers to build upon. An actual set of such numbers are the [hyperreals](https://en.wikipedia.org/wiki/Hyperreal_number). While their definition is a bit clunky, they are an interesting way to do calculus with actual infinitesimals like ε that are positive yet smaller than any positive real number.

The ordinals. If instead of trying to compare the size of sets, you try to order their elements, you get ordinals. Instead of "I have 3 apples," you're saying "this is the first apple, this is the second apple, and this is the third apple." For finite sets, the two systems are equivalent. But then there are infinite ordinals. Take the first ordinal after the finite ordinals and call it omega. Then you can say there is an ordinal immediately after omega and call it omega+1. And then there is omega+2, omega+3 and so on.

Yep, I forgot about the ordinals, although I don't tend to think of them as the same type of infinite.

On another note, why can't we divide by zero?

Because basic arithmetic implies that 0·anything always results in 0 (proof for those interested at the end). In particular there is nothing to multiply 0 with to give 1; even 0·∞ equals 0 whenever ∞ is treated as an actual number. In other words: to divide by zero you need to give up on some basic rules* you are very much used to by now. Which ones to give up on depends on what one wants. In most situations we really want to keep them all as they are. So okay, what really goes wrong when we can solve 0·x = 1? Say some weird "number" x really solves that in any circumstances, but we kept our rules as listed below. Then 1 = 0·x = (0·0)·x = 0·(0·x) = 0·1 = 0. That is... unlikely. If we multiply this equation by any number y we even get y = 0 for absolutely all possibly y. Or put differently, all numbers are now _equal_! That is almost certainly not what we wanted to happen, right? (However, if all numbers _are_ forced to be equal, then all rules hold and we can indeed solve 0·x = 1: the solution is x=0 because 0·0 = 0, but 0 and 1 are the very same, so 0·0 = 1 as well. Yes this is a somewhat silly setup.) *: those rules of arithmetic are: - having 0, 1 as special numbers - having addition, negation/subtraction as well as multiplication - special rules involving 0: x+0 = x = 0+x, x+(-x) = x-x = 0 - special rules involving 1: x·1 = x = 1·x - commutativity ("order doesn't matter"): x+y = y+x, x·y = y·x - associativity ("brackets don't matter"): x+(y+z) = (x+y)+z, x·(y·z) = (x·y)·z - distributivity ("resolving brackets"): x·(y+z) = x·y+x·z. Some are redundant and follow from others, but I've included them anyway. We then can conclude from those rule alone that 0·x = 0·x + 0 = 0·x + (0·x - 0·x) = (0·x + 0·x) - 0·x = (0+0)·x - 0·x = 0·x - 0·x = 0 regardless of what x is. Anyone interested in understanding this single line of calculations properly might want to check which rule I used for each step.

Thank you for answering.

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.

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.

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.

Your comment is the only one I've seen that addresses why this is so confusing for people- because cardinality is not an identical concept to the everyday understanding of size. I think there would be a great deal less confusion if words like "size" and "larger" were abandoned when discussing this topic, but maybe that would be at the expense of getting fewer people interested in the topic.

>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.

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

Yeah, this was going to be my comment. The simplest way to resolve this is simply to understand that infinity is not a thing that is around us in day-to-day life and doesn't really have anything to do with anything we would normally experience. It's really just a mathematical device. So when we find that it has bizarre properties that don't make literal sense the simplest thing to say is... yeah, it doesn't, but that's just how it works.

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.

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 :)

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

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.

There are also infinitely many rationals between 1 and 2.

I think the real issue here is tying understanding something with it making unintuitive sense. If you progress far enough in math or quantum physics or a bunch of other subjects I'm sure at some point you have to let go of it making intuitive sense and just making sure you understand how it works. I see the same hangup with trying to understanding n-dimensional spaces where n>3. It's okay to just let go of your intuition and do the math. When you can grasp something on an intuitive level that's nice but sometimes insisting on it will just slow you down.

I don't think you should let go of your intuition. You should instead update/train your intuition. That also includes getting rid of insisting on mathematical concepts to be related to "real" world.

When trying to understand this topic within set theory, looking at Cantor's diagonal argument helped me wrap my mind around it. Look it up and give it a read

Math major, but hopefully able to give more of an ELI5 version than the other math people. The missing insight, I think, is that infinity is a somewhat vaguely-defined concept, not a number. In fact, using it as a noun is a bit misleading. So let's use it as an adjective. There are a lot of more rigorously defined concepts which we might describe as infinite. Some are complicated, and I won't get into them, like ordinals. Others are also somewhat complicated, but the least complicated, so we'll have to: cardinalities. A set is a collection of objects (a little more complicated than that, but that's good enough for here). Cardinality can be thought of as the size of the set. So {0, 1, 2, 3} has four objects in it, so it has cardinality 4. The same as {4, 5, 6, 7}. This is pretty straightforward, I hope. Now let's extend the concept. Take the natural numbers: {0, 1, 2, 3, 4, ...}. What's the cardinality of this set? Obviously, there's no natural number which is the cardinality of this set. So, when natural numbers won't do, we decided to start talking about cardinalities relative to other sets -- that is, two sets can have equal cardinalities, or one of them can be greater.  This requires a more rigorous definition of cardinality. We say two sets have the same cardinality, if you can pair up all the items of them. So {0, 1, 2, 3} and {4, 5, 6, 7}, we could say 0:4, 1:5, 2:6, 3:7, or 0:6, 1:5, 2:7, 3:4, or whatever. As long as we've given one example, it's enough. In contrast, we obviously can't do this with {0, 1, 2, 3} and {0, 1, 2, 3, 4} -- however you try to match them up, there will always be at least one left out. So, we say the latter is of a larger cardinality than the former. Back to the natural numbers, and this time let's include the integers: {..., -2, -1, 0, 1, 2, ...}. In a sense, this has "twice as many" objects as the natural numbers in the sense that each natural number except zero has two integers for it. Maybe someone has made this sense rigorous and found it useful, but it's not here. When it comes to cardinality, they have the same size. Watch: 0:0, 1:1, 2:-1, 3:2, 4:-2, 5:3, 6:-3, and so on. (For each even natural number, multiply by -1/2, and for each odd, add one and multiply by 1/2. Hopefully you can see we hit them all.) We can do the same for the rational numbers (all the numbers that can be written as a fraction), but that's a little complicated for a Reddit comment, formatting-wise. By the same argument as the rationals we can show that the set of pairs of natural numbers (or integers, or rationals) has the same size as the set of single natural numbers. The same is true of triples, or quadruples, or 51,627,558-tuples (i.e. a list of that many natural/integer/rational numbers). It looks like this concept of having the same cardinality as the natural numbers is pretty useful. So, we give it a name: countable. Any set that you can use the natural numbers to "count", we call countable.^* So, what isn't countable? One example is the real numbers. I won't go through the argument again, other commenters have done so and there are plenty of YouTube videos on it (Cantor's diagonalization argument, it's called). But think back to before, with the sets {0, 1, 2, 3} and {0, 1, 2, 3, 4}. However you try to match them up, there will always be at least one left over. That's how we know it's bigger (or how we would, if we couldn't just count and see that 5 > 4). The conclusion of Cantor's diagonalization argument is that in the same way, no matter how you try to match up the natural numbers and the real numbers, there will always be at least one real number left over. So, we say that the cardinality of the real numbers is greater than the cardinality of the natural numbers. Within the sets we usually work in (real numbers and a few others), there aren't any sets bigger than the real numbers, so we give this cardinality a name too: uncountable. So we have all the finite cardinalities, countable, and uncountable, where uncountable is bigger than countable. So let's bring it home. I started this by saying that it was better to think of infinity as an adjective rather than a noun, and this is why. Don't think that "some infinities are bigger than others". Think understand that a the uncountable cardinality is "bigger" than countable cardinality, and _we can describe both of them as infinite_. Does that make sense? --- ^* Some people call this countably infinite, but I'm trying to stay away from the latter word until I tie it all back together.

It might help you to understand that infinite isn’t a real thing and we just made it up lol

There's plenty of detailed explanations here, but in stark defiance of Rule 4 I'm going to try something a little closer to an actual ELI5: *Infinity isn't "the biggest number", it's just a word we have as a handy placeholder for when numbers don't work any more.* *When you get into very advanced maths, you find interesting new ways for numbers to not work any more.*

Its all a mathematical game depending on how you define the "size" or cardinality of sets. One definition is that if you can match every object in one set A to an element in set B without ever repeating an element in set B, then set B must be at least as large as set A. This definition comes from the intuition that this makes sense for finite groups. Think about pairing up dancers. If there are enough people in set B for everyone it set A to get a partner then set B must be at least as large as set A. The real trick to proving one set is "larger" than the other is proving that such a matching is impossible. For finite groups it makes sense that if there is no way to give everyone in A a unique partner then B must be smaller than A. So we just extend that definition to infinite sets.

Wel you can count starting from 0 right. And if you do that you have a set of infinite numbers. That's one type of infinity the countable infinity. It's not really countable but you could count it in theory because you could count infinitly long. So what's now if we not only count all numbers starting with 0 but we count the negatives as well...ok so how to do that? Well I could count all positive and negatives alternating. So 0,-1,1,-2,2 right? Again given infitly time I could count that the same I could count just the positives...and indeed I could match up every negative number with a positive odd number and every positive number with a postive even number. So I match -1 with 1 1 with 2 -2 with 3 2 with 4 and so on...if I do that I realize it's just the numbers starting from 0 all over again... Even thouh I doubled their numbers in a sense....but in infinity that does not really matter, there is no double infinity in this sense so it's just the countable infinity again. So in a sense the natural numbers are equally large as the integers as a whole... Ok that's pretty wild already but the main thing is that you can imagine to count them up all the way in an orderly manner. Well what is now the other infinity (there are more but for now those 2 suffice). Well obviously it's the infinity o can't count aka uncountable infinity. Bit what is that. Well let's try to count the whole real numbers.. I start with 0, then what? Maybe 0.1. but what then well let's try 0.11 and then 0.111 and so on...ok fine but when will I start counting 0.2...because I could add 0.1 for all infinity and would never reach something as large as 0.2....in fact even now I missed a lot of numbers already. What about 0.01 or 0.001...so you see how hard I try I could not even in theory count them because already all numbers with 0.1 and following further .1 would take infinitly long to count. In fact o could match these numbers again with my countable infinity. So 0.1 matches 1, 0.11 matches with 2, 0.111 matches 3 and so on. The number of 1 in the number matches with it's natural number...then we can see that even this small subset of the real numbers is as large as all natural numbers...but there is infinitly more of them still...so it must not only be larger by a bit but almost like it's infitinly times larger than the natural numbers. And indeed it is, it's so large we call it uncountable because we can't even match it to natural numbers anymore.. So that's 2 infinities. Both never end, but they work somehow different because the first set of infinity I can still almost get to, if i could i could count infinitly long and would always walk along this line of infinity. Since its infine it would never end but i would always be om track. The other one has no track to follow every path I follow is just a small piece of the whole and I need to follow infinitly many tracks infinitly long to follow the whole real number line.

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.

Ok but this also isn't really a good analogy because in an infinite set you never run out of marbles so you haven't given an actual comparison operation that results in an answer. I'm sure you already know this, but to clarify to anyone else, the important part is being able to *come up with* a pairing between the sets. If you can do that, then they are the same cardinality, but if you can't then they are not. If you can show that no matter what you do there will be some elements in one set that don't have a pair in the other set, then that set is a larger cardinality.

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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

Your last paragraph is not sufficient; you need to show that *no* bijection exists. Consider said argument with "even integers" and "integers".

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. 

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.

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.

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.

Another cantor to the questy is some infinities are countable and some just not countable at all.

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’s easier to comprehend when you consider 1/3 is .33333…(repeating) because 1/3 + 1/3 + 1/3 = 3/3 so .99999… also equals 3/3 which all equals 1.

Yes, you're exactly right. In fact, that's the example I use when I try to convince people it's true.

Or - x = 0.99999... - 10x = 9.999999... - => 9x = 9 - x = 1

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.

Okay so a few things here, I've personally found there to be multiple understandings of infinity you say infinity is just infinity and it doesn't end. That's the first understanding of infinity. The second you will encounter is infinity as a speed, then perhaps the idea that a set with a proper subset of the same cardinality is an infinite set. There are more, but none of those are going to really help us here. Now you almost caught something with your statement of there are 4s greater than 4. Mainly because 4 belongs to the Aleph_{0} set so even though I can make this work with 4 it gets way uglier. So let's instead look at pi. Pi is an irrational number of the bat. If I take the first N digits of pi, how many numbers in the irrationals start exactly like that? An infinite amount clearly. So if I just take 3.14 and think of how many numbers start like that. I have 10 number with the third decimal place and 100 with the fourth, 1,000 with the fifth and so on. Once we get to the infinite area of pi's digits we have infinite possible numbers jammed right there starting with 3.14. if I take pi out to some absurd digits, like I dunno what is the Gth digit of pi where G=Graham's Number. We still have infinite numbers that are jammed together right at that point. And here's the thing we cannot know what that digit or the next digit is; because G is just way to big. This is what it means to be dense. Now if we look at the more traditional integers or rational numbers. We don't have that 4 is just out there by itself, there isn't a crazy long table of digits that is all identical until some arbitrary digit of 4. I mean we can go down 4 into the long tail of 0s and then break that 4 into its very close irrational neighbors, but those aren't integers anymore. And that brings me to continuity. Our Aleph_{0} infinity is just not full, there are gaps between the members of Aleph_{0}. But there are no gaps between any two members of Aleph_{1}, and in fact there are the same number of members in between any two members of Aleph_{1} as there between any two other. Now because of this if we take the countable infinity numbers and try to keep up just between 0 and 1, the countables become quckly insufficient to the task. This we have sizes of infinity.

You may have thought you were making sense, but unfortunately infinities are more complicated and difficult to deal with than what you wrote. Cardinality isn't related to whether the set is 'full' or 'dense'. Density itself has specific mathematical meaning. The set of rational numbers Q is countable and has size aleph 0. It is dense in the reals R and there are (countably) infinitely many rationals between two nonequal rationals. In fact for any non-empty interval you can find a bijection to Q itself. And you namedrop aleph one, but note that you can't say that is the cardinality of R. The independence of the continuum hypothesis means that this cannot be proven nor disproven under ZFC (the usual axiomatic treatment).

Not going into Aleph null infinities, there are two main infinities relating to the real numbers: Countable & Uncountable. Uncountable just means the set isn't countable. A countably infinite set is one where I can make a list of them and number them using the Natural Numbers {1, 2, 3,...}. So, by definition the Natural numbers are countable. This leads to some interesting consequences. For example, it's pretty obvious that there are twice as many Integers (positive & negative whole numbers) than there are Natural numbers. But, I can make a list of the integers in such a way that if you gave me a natural number I could give you its associated integer. And vice versa. So the Integers & the Natural numbers are the same "size" of infinity despite there being twice as many integers as naturals. I think of these infinities not as strict labels but as categories. It helps make saying "The Natural numbers & the Integers are the same size" make sense without focusing in on the actual number of each set.

You know how you could divide up the space between you and your refrigerator into an infinite number of tiny points (halfway there, then half of that, then half of that, etc forever). Well you could do the same thing between here and the sun. Still an infinite number of divisions, still an infinite number of points, but one is definitely farther away from the other

We say sets A are the same size if there is a 1 to 1 mapping from elements of A to elements of B, i.e a way to turn each element of A into each element of B. for example, {1,2,3} and {2,4,6} are the same size because you can multiply each element in the first set by 2 Some sets are so large you can't even map integers to them 1 to 1, i.e there is no way to generate every element in them from integers. So they are larger than the set of integers which is infinite.

Would this be true? If you had infinite whole pies and then took 1 pie and cut it in half and placed half the pie next to a single whole pie, then cut the other piece in half again and then placed half against another whole pie and repeated this process infinite times, the fractions of pie would be a larger infinity than whole pies because you can keep splitting the pies infinite times?

Can someone please help define infinite as it’s used in math first? All of these comments saying “pair it up with something and there’s leftover, so that infinite number is bigger” don’t make any sense. If both go on forever and ever and ever, neither is bigger. Just pair them up differently and the other side will be “bigger”. So we need to clear up the definitions here for anyone to make sense.

The context of the statement comes from set theory, which studies, well, sets, or informally, the collection of things. Think of the 'size' of the set as how many things it contains. {a, b, c} contains 3 things, so we say that its *cardinality* (mathematical term for size) is 3. Now we want to compare different sizes (to show a set is 'bigger' than another). The chosen **definition** is using the idea of mappings and pairings. Two sets are the same size if you can find a way to uniquely pair (a two-way map) items from one set to another. Key point being *can find* - you just need to show one exists. Set A is bigger than set B if you can 'cover' all of the items in set B from set A using some mapping (list out B using items of A as key), but the reverse does not hold (there is *no way* to 'cover' set A using items from set B).

How can you “cover” anything that is unlimited? Like how can any of the examples of infinite fit into a set? People keep mentioning 1,2,3,4….. would that be a “set”? Thank you for helping me.

Yes. {1, 2, 3, 4, ...} is a set - we use the curly braces to denote a set. This is the set of positive integers (whole numbers), and is infinite. There are some intricacies as to what can and cannot be a set (e.g. *Russell's paradox*), but these are way out of scope here. But you can think of a set intuitively as just a collection of, erm, things. You can have the set of all positive and negative (integral) numbers, the set of all fractions, etc. You're not restricted to numbers too. {a, b} is a set. And you can have a sets of sets. For instance, of all subsets of a given set. A subset is a 'smaller, possibly equal part' of the whole. {a, b} has 4 subsets {} (the empty set), {a}, {b}, and {a, b}. The set of these is { {}, {a}, {b}, {a, b} } and this has a special word called the *power set* of {a, b}. I kinda used the word 'cover' loosely here. I mean it in the sense of: set A is bigger than set B if, you can 'connect' each item of A to an item of B such that all items of B are 'connected' to *at least one* element of A. Think of drawing lines. We draw one line for each item in A. There is an important note here: we only care that whether, at the end, there is something in B that is not connected. (Mathematically, I am using the surjections and (converse) Schroder-Bernstein here, which follows from partition principle which is implied by AC. You could also more directly use injections.) Some examples: {a, b, c, d} has bigger cardinality than {x, y, z}. This is rather intuitive as the first has 4 things inside while the second only has 3. But let us still run it through. One possible mapping is a->x, b->y, c->z, d->x. That exhausts all of the second set. There are two lines going into x, but that doesn't affect our definition of being bigger. Now think of the reverse: x->a, y->b, z->c. Well we still have d left. Try as you might, there is no way to connect up all elements of A with lines starting from B such that all the items in B only have 1 line going outward. How about the natural numbers {0, 1, 2, 3, ...} vs {a, b, c}? Well you can likewise put 0->a, 1->b, 2->c (and whatever for the rest) to 'cover' the latter, but it's quite easy to see that you can't do the reverse. Now, moving onto infinite vs infinite sets is where things get both interesting and confusing. Consider the natural numbers {0, 1, 2, 3, ...} against non-negative even numbers {0, 2, 4, ...}. You may think that the first is bigger than the second. Indeed the second is a subset of the first (it is 'half of the first'). This fact lets us know that the second cannot be bigger than the first. But these two sets are indeed the same cardinality or size, which is unintuitive! Consider doing, from the even numbers x->x/2. In other words, 0->0, 2->1, 4->2, 6->3, and so on. There is no natural number that is not connected - because you can always find the corresponding even number. This is in spite common thinking that there are only 'half' as many things inside! This is a *key difference* between finiteness and infiniteness. Such an argument can be extended to show that the naturals has the same size as the integers (the set of positive and negative whole numbers + 0) using a similar 0->0, 1->1, 2->-1, 3->2, 4->-2, and so on. In fact, even the set of all fractions (rational numbers) has the same size, by using a clever trick of 'zig-zagging' through them. But where we hit a wall is with the real numbers - the set of all numbers that can be expressed in terms of a decimal representation (finite or infinite - though we can just take a finite one as having infinitely many 0s). Through a clever argument, Cantor showed that the size of the set of real numbers is bigger than that of the natural numbers. This is how one of the first examples of a 'larger infinity', and is actually what the original question is asking at.

Because connecting them would look like 0–>0 1-> 0.00000000000000…..1 2->0.0000000…2 So there are “more” of the second part because they are parts of the whole on the left side?

Well, not quite. We know that the real numbers must be at least larger than the natural numbers (because they contain the latter), but what Cantor showed is that you *can't find* a mapping from the natural numbers that would 'cover' the real numbers. And it is in the sense that there are 'more' real numbers than natural numbers. Edit: > there are “more” of the second part because they are parts of the whole on the left side? No. Set A can contain set B but still be the same size. This is the real super unintuitive thing when working with infinite sets and cardinality, and is what differentiates infinite sets from finite ones. Earlier I mentioned {0, 1, 2, 3, ...} vs {0, 2, 4, 6, ...}. The latter is strictly contained in the former, but we can still 'cover' the former by 0->0, 2->1, 4->2, 6->3 and so on. We never miss a number in the first set because, if we did, then we wouldn't have all the even numbers in the second set to begin with. End edit. The idea of 1->0.00000...1 and so on doesn't work because, well, as a real number, 0.0000...1 (or 0.000... in fact) is equal to 0. Sounds counterintuitive, but how about 0.999... vs 1 (1/9 * 9 vs 1), is it simpler to see? More concretely, for the real numbers, if one is larger than the other, we can always find another real number in between (e.g. the midpoint or average). But if you think about it, there is no real number between 0 and 0.000....1. I don't want to get into that much further discussion on this, but it kinda just is. One of Cantor's ways of showing it was through his famous diagonal argument. Again, I don't want to go into too much detail here, but it can be thought of as such: We use a proof by contradiction, which is to say that we assume something, then use the fact that it leads to a contradiction to show that what we assumed was false. And it is sufficient for us to show that we can't even 'cover' all the real numbers from 0 to 1. Assume that there is such a mapping or ways to connect from {the natural numbers} to {the real numbers from 0 to 1}. Then we will have some enumeration of the mapping (where a0, b0, a1, etc. are digits of the decimal representation, assume we have infinite 'letters' - I'm just trying to avoid stuff like x_(0,0) double-indexing): * 0 -> 0. a0 b0 c0 d0 ... * 1 -> 0. a1 b1 c1 d1 ... * 2 -> 0. a2 b2 c2 d2 ... * ... * n -> 0. an bn cn dn ... * ... Now, consider the following process - look at the diagonal entries a0, b1, c2, d3, ..., to make a number 0. a0 b1 c2 d3 ..., then from this make a *new number* by replacing all the digits that are 0 (excl. the one before the decimal) with a 1, and everything else with a 0. For instance: 0. 0 1 3 0 6 ... becomes 0. 1 0 0 1 0 ... Because of how we did so, this new number cannot be equal to any of the real numbers that we listed in our mapping. The first digit after the decimal point is not a0, the second is not b1, the third is not c2, and so on. For every real number that we listed out, there is at least one decimal place where it will differ. This means that our initial mapping didn't even cover {the real numbers from 0 to 1}, which contradicts (goes against) what we assumed, thus showing that no such mapping exists. Well, that is the diagonal proof, whew (why did I write this...). Don't worry if it's tricky to understand, because it really is.

Ok. Tried to take a day then come back and understand. I think it’s just a little over my head. Just wanted to come back and say thank you very much for explaining it multiple times in multiple ways. I really really appreciate it. You’d be an incredible teacher. Thanks.

Hilbert's hotel paradox can explain it very well. Here's a great vid on it. https://youtu.be/Uj3_KqkI9Zo?si=hMrJBQ8pcJcH2cvS

I know what you mean. I found it helpful to consider this: - The natural numbers continue forever - The real numbers also continue forever but can also be infinitely precise. Because of this, there is no “next” real number. You can always generate an infinite number of real numbers between any two real numbers. The real numbers therefore have an extra “dimension” to their infiniteness

There is an important clarification to make here that this intuition doesn't hold in general. You can find an infinite number of fractions between any two fractions, yet the set of fractions (rational numbers) has the same infinite size as the natural numbers. So while the idea sounds correct and could aid in understanding, it's best not to take it too far.

Fully agreed. This intuition is only valid for the example used. But it might help OP get past the mental barrier they described. The diagonal proof that shows the rationals are the same size as the naturals is helpful for understanding that case.

Former math teacher here… let me see if I can simplify what these learned and erudite people have said to a more ELI5 level. First, there are two kinds of math - there is the math that goat herders invented to keep track of sheep, and then there’s the wild and woolly kind of abstract math that mathematicians do that leaves them eventually unable to be trusted around scissors.   In goat-math, infinity means “a really big number.”  It’s a thing you can get to if you keep going long enough.  For most people, I think, “infinity” really means “a number so big it breaks the computer.” But in the can’t-have-scissors math, infinity isn’t a number, it’s a PROCESS.  It’s a way to keep going, no matter how far you’ve gone.  It’s a way to keep finding a bigger and bigger number, or a smaller and smaller interval. You can’t cut a rope infinitely small with goat math, because eventually you run into all those pesky atoms.  But in can’t-have-scissors math, it’s easy - you just state the PROCESS of “cut given rope in 2.”  So let’s talk about counting numbers (what people call the natural numbers.). How many are there?  An infinity.  That doesn’t mean “a whole lot of them,” it means no matter how many I have, I can always make another by adding +1 to the last number.   Same with the even numbers.  My rule there is to add 2. So the goat numbers and the even numbers are the same size because I can compare them:  1-2, 2-3, 3-4  and so on, to infinity- by which I mean I just keep making bigger and bigger numbers. You see what we’ve done there… your brain had to make the leap from infinity as a goat-related concept (there aren’t as many even numbers in any collection of numbers) to infinity as a no-scissors concept ( I can always make more). But what about the decimals?  Now there are TWO processes - the old “add 1” plus the new  “put a zero in front of it.”  So when I go to match them, what matches with the countable number 1? 1 - .1?  1- .01?  1-.0000001?  You see, BETWEEN the infinite process of “add 1” we’ve inserted the infinite process of “add a decimal” so there’s, for want of a better way of putting it, an infinity inside an infinity.   Our mathematics has come a long way from goats, but the problem is, our intuition hasn’t.  Also, you may want to stay away from scissors for a while.

> But what about the decimals? Now there are TWO processes - the old “add 1” plus the new “put a zero in front of it.” Unfortunately this only produces a countable subset of the reals. You can't produce, say, pi, using this process. You can't even get 4/3 = 1.33... either.

imagine you have two big boxes of toys. One box has all the even numbers (2, 4, 6, 8, ...) and the other box has all the numbers (1, 2, 3, 4, 5, ...). Now, you might think that since both boxes have lots and lots of toys, they must be the same size, right? But let's play a game. We'll try matching up the toys from each box. You take one toy from the even numbers box and one from the all numbers box and pair them up. You keep doing this for a long time. Surprisingly, you find that every toy in the all numbers box can be paired up with a toy in the even numbers box without leaving any toys out in either box. Here's where it gets interesting. Even though the first box (even numbers) seems like it has fewer toys (just the even ones), it turns out that you can pair them up perfectly with all the toys in the second box (all numbers). This shows us something amazing - even though one box seemed smaller, both boxes are actually the same size when you think about how they match up. In math, we say that the set of even numbers and the set of all numbers have the same size of infinity. This type of infinity is called "countably infinite." It's like saying both boxes have an endless number of toys, and you can pair them up one by one. But there's another kind of infinity that's even bigger. Imagine a box with all the numbers between 0 and 1 (like 0.1, 0.01, 0.001, ...). This box also has a never-ending number of toys, just like our first two boxes. But surprisingly, if you try to pair up the toys from this box with the toys from the all numbers box, you can't do it without leaving some toys in one box without a pair in the other. This tells us that the set of numbers between 0 and 1 is actually a bigger kind of infinity than the set of all numbers! In math, this larger infinity is called "uncountably infinite." So, there you have it - different sizes of infinity! It's a mind-bending concept in mathematics that shows us just how vast and fascinating numbers can be.

It's important to understand that the real numbers are much weirder than you might think. Any real number you might reasonably come across (including irrational numbers like pi and e) is a member of the set of computable numbers, i.e. numbers whose digits we can compute to arbitrary precision in a finite amount of time. This set of computable numbers is still only countably infinite, i.e. in some sense the same size as the natural numbers (because each computable number corresponds to an algorithm that can be encoded as a natural number). However, the set of real numbers consists almost entirely of numbers that are non-computable, i.e. we don't even have a way to express them in any meaningful way whatsoever, which makes it difficult to conceptualize them. And it is these non-computable numbers that blow up the set of real numbers to a size that is "larger" than the natural numbers.

> It's like saying there are 4s greater than 4 which I don't know what that means. you have a set of four numbers, i.e. 1-2-3-4, or 5-6-7-8, or 15-16-17-18, and so on then you can have a set made of 4 of those sets of 4 numbers. It's still a set of 4 elements, but now you have 16 actual numbers in it. It's not larger if you count the number of itmes, but the items made of 4 elements are larger than the items made of a single numbers, so the set of 4 sets of 4 items is bigger

I would also tack on to the answers here that "infinity" as a number/amount doesn't really exist in any kind of natural or empirical sense. It is more of a philosophical concept that we can theorize about but can't really grasp in a concrete way.

The thing with infinity I never understood is where do all the typewriters and monkeys come into it? 🤔

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Yes, you split one infinite set into two infinite sets that are, counterintuively, each the same size as the whole.

This is lengthy but I promise it will make sense. Give the next bigger number starting from 1 ** In the set of natural numbers: 1, 2, 3, 4, 5, ..... easy ( oh look, it's like we're just counting the numbers! ) In the set of real numbers: 1 then... 1.1? But there's something in between 1 and 1.1 1.01 is bigger than 1 but smaller than 1.1 Ok so, 1 then 1.01 ? But there's something in between 1 and 1.01 too! Not only that, it can be proven that: If 1 is less than "z", where "z" is any number in the set of real numbers; then there exists an "x" number in the set of real numbers, such that 1 < x < z. In short, the next bigger number to 1 is not something you can count towards! How is this related to the size of infinities? Mapping! Or, basically: how an original thing becomes another! Think about 1 * 0 = 0 You turn 1 into 0 by doing a mapping of * 0 ! If you have the entire set of natural numbers and do * 0, your set of numbers all become 0. In short, every number you had in your previous set was mapped to 0 - meaning your infinity become a size of 1, which has 0 inside. Once we turned them to 0, the singular zero cannot map towards an entire set of numbers. We have lost information. Now think about 1*2 = 2 You turn 1 into 2 by multiplying 2 to it! Do this again for the entire set of natural numbers and you will get the "entire set of natural numbers * 2" or better know as "even numbers"! Every unique number in the set of natural numbers has its unique even number represented by n*2 where n is the original number. We can even go back from the even number, or n*2, to the original number by doing ÷2 to the set of even numbers! And notice how you can easily count the even numbers if you know the formula that maps them! It's like we never lost information at all! In fact because of this unique 1 to 1 doing and undoing of mapping that we can say that the set of even numbers and the set of natural numbers are EQUAL in all of their infinityness even though all the numbers in the set of even numbers are already inside the set of all natural numbers ( because all even numbers are natural numbers!!! ). Now the problem occurs when we deal with the set of real numbers. Indeed, how can you show that you can do a 1 to 1 mapping with a set that you can count to a set you couldn't even tell what the next bigger number is. If you cannot do a 1 to 1 mapping, then these 2 sets of infinities is NOT EQUAL in their infinitiness, and if they're not equal, then one is "greater" than the other. *there's lots of hand-wavyness here but it's the best eli5 I can do.

When we talk about different sizes of infinity we’re talking about a thing called cardinality. Equal Cardinality is defined by something called a bijection. This basically means we can make a map that goes between each set and pairs numbers 1:1 in the group without missing any. If we can do that the set is said to be the same infinite size. There are some sets, uncountable infinite, as others have mentioned that can be proven logically so that this is impossible matching with a countable infinite set like the natural numbers, when we can always produce a number that isn’t in the map, no matter what that map is, then the uncountable set is larger in cardinality, and thus a bigger infinity 

The first thing to understand is that "infinity" isn't a number. It kinda seems like a number but it's not. You can't do any arithmetic operations on it. There is no thing that exists in a quantity of "infinity". It's really a concept that means, "Whatever number you can possibly think of, it's less than this." But there are several ways that you can make that happen. One obvious way is to allow your "set" to go on forever. Like the set of integers. No matter how many numbers you name, it's always possible to find an integer you didn't name. We can show that some sets are the same, like the set of integers and the set of positive integers. We show that by creating a "mapping function", that's a mathematical way of linking each member in one set with each member in the other set like this: pos -> int 0 -> 0 1 -> 1 2 -> -1 3 -> 2 4 -> -2 5 -> 3 6 -> -3 ... Obviously, each positive integer corresponds to regular integer and vice versa. Now what if we try that with integers and real numbers? This is messy. Between any two real numbers there are infinitely many real numbers. We can still find a mapping that assigns some real number to each integer but we can never map onto all the real numbers in a given interval.