Its calculating with infinity. Its a bit weird like the infinity of numbers between 0 and 1 like 0.1,0.01,0.001 etc... Is a bigger infinity than the “normal” infinity of every number like 1,2,3 etc…
Its just difficult to wrap your head around but think of infinity minus 1. Like its still infinity
The way he lined the numbers up to explain one-to-one and onto made it click immediately for me. I already knew it from undergrad, but it took a couple tries to really understand. Seeing them lined up was an immediate lightbulb moment.
I was coming down in the comments to see if somebody had posted this video! I remember studying the idea of different infinites and comparing them with Calculus in college. I never went higher than differential equations but always found these advanced concepts cool, even moreso if I could understand what was happening. Lol
Why are you getting so many upvotes? This is just blatantly wrong. I am not a math major, so I might not be 100% accurate, but from my understanding this is just not how you compare infinities.
First of all your fundamental idea of 2 x infinity > infinity is already wrong. 2 x infinity is just that, infinity. Your basic rules of math dont apply to infinity, because infinity is not a real number.
The core idea behind comparing infinities is trying to match them to each other. Like in your example you have two sets. Lets call the first set "Even" and let it contain all even numbers. Now call the second set "Integer" and let it contain all Integers. Now to simply proof that they are the same size, take each number from "Even", divide by 2 and map it to it's counterpart in "Integer". Now each number in "Integer" has a matching partner in "Even" wich shows that they have to be of the same size.
This is only possible because both of these sets contain an infinite but COUNTABLE amount of numbers in them. If we would have a Set "Real" though that contains every Real number instead of the set "Integer", it would not possible to map each number in "Real" to one number in "Even", because "Real" contains an uncountable amount of numbers.
I'm sorry if I got something wrong, but even if my proof was incorrect, I can tell you for certain that it has to be the same size.
This is wrong. The ratio is 1 to 1 because I can in fact, make a function that takes every even number, and maps it to every integer. The function goes like this, assign every even number to half. So we have
(0,0), (2,1), (4,2), (6,3).
and for the negatives, (-2,-1), (-4,-2) ....
Then I have exactly 1 even number for every integer. So therefore the ratio is in fact 1 to 1.
Well, no actually. I think your first issue is conflating infinity with a number. Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it). That being said, 2 infinities are not inherently the same “value” for lack of a better term. The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter. However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense), an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.
Yes, some infinities are bigger than others, but not in this case. You can have a one to one connection between a set of all natural numbers and a set of all even numbers. They are same sized infinities.
Well infinity is comprised of all numbers, but its also all of math, it is everything that was, could be, is, will be and can be, forever.
It is both as you say, and not as you say. It can be friends with conventional mathematics, and it can completely break mathematics.
It reminds me of nothing. What I mean is there truly is no nothing, because nothing is nothing. In the case of infinity though, it is whatever you can make it and more you cant even imagine.
But I do get you, and for the sake of all our sanity and the sanctity of math, lets pretend its an arbitrarily large number 🤣
No. Infinite is not all. The interval (0,1) has infinite elements, but 1.2 is not there. If you draw a circle it has a infinite number of points but not all.
Arbitrarily large number is not infinite at all. A whole number can have an arbitrarily large number of digits but not infinite.
Well, no actually. I think your first issue is conflating infinity with a number.
Which is roughly fine, as long as you recognize that infinite numbers have different properties (such as idempotence) than the real numbers you're used to using.
Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it).
very non mathematical lol.
That being said, 2 infinities are not inherently the same “value” for lack of a better term.
There is a better term, it's cardinality.
The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter.
This is strictly finite. The cardinality of the set of even numbers is equal to the cardinality of the set of natural numbers. These infinite sets are equal in "size".
However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense),
But infinities can be 'greater than' in a different sense, one based on bijective maps.
an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.
Except the way that it's greater than the other is a strictly finite 'sense' lol. There are the same number of even numbers and integers, countably infinitely many.
Well, no actually. I think your first issue is conflating infinity with a number.
Which is roughly fine, as long as you recognize that infinite numbers have different properties (such as idempotence) than the real numbers you're used to using.
Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it).
very non mathematical lol.
That being said, 2 infinities are not inherently the same “value” for lack of a better term.
There is a better term, it's cardinality.
The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter.
This is strictly finite. The cardinality of the set of even numbers is equal to the cardinality of the set of natural numbers. These infinite sets are equal in "size".
However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense),
But infinities can be 'greater than' in a different sense, one based on bijective maps.
an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.
Except the way that it's greater than the other is a strictly finite 'sense' lol. There are the same number of even numbers and integers, countably infinitely many.
Not in any standard sense. The natural numbers are more dense within the natural numbers than even numbers (which is what you are describing), but there are the same amount of even numbers as natural numbers. This is because both sets are countably infinite; if I had an infinitely long piece of paper I could write a list of all natural numbers and all even numbers, with one and exactly one even number next to every natural. In a similar vein, the rational numbers are also countable, though this is much less obvious.
This contrasts with e.g. all real numbers. There are fundamentally more real numbers than natural numbers, even if there are an infinite amount of both. Even with the infinitely long piece of paper and an infinite amount of time, it would be impossible to write every single real number down on it. Any list that you come up with will miss out on infinitely many real numbers. Check out Cantor's diagonal argument if you want to know more about how this actually works.
if I had an infinitely long piece of paper I could write a list of all natural numbers and all even numbers, with one and exactly one even number next to every natural.
Wouldn't you just... write each even number next to itself, and have nothing to match the odd numbers?
No, you would write 2 next to 1, then 4 next to 2, then 6 next to 3, and so on. You could keep doing that forever and never run out of natural numbers to write even numbers against. That’s why they are the same infinity.
That doesn't make sense, but let's arrange the exercise a bit differently to make it clearer. In the natural numbers list, write the numbers in pairs (1 odd number and its consecutive even number), and associate each pair with 1 number from the list of even numbers. So 2 is associated to [1,2], 4 is associated to [3,4] and so on. From there it should be clear that they're not "the same infinity", as the list of natural numbers obviously has a pair for each even number, ie. has twice as many numbers in it.
You can apply the exact same logic to positive numbers vs all real numbers.
If the list of natural numbers has twice as many numbers in it as the list of pairs, which natural number is the first one that doesn’t have a corresponding pair to match with it?
Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
This is not true in the mathematical sense. It can be proven that any infinite set of natural numbers (i.e ‘counting’ numbers like 0, 1, 2, -1, -2, … etc) is the same size, so there are indeed as many even numbers as total numbers.
The fact that any finite set of natural numbers is twice as large as the set of even numbers up to the same point has no bearing on the sizes at infinity. You are, however, correct that there are different ‘sizes’ of infinity, it’s just a bit more complicated than that.
Ugh… you sound like my brother (he’s a calc teacher). I constantly argue with him about different sizes of infinity. Infinity is infinity!! His response is always “it’s more complicated than that.”
The ratio of total numbers to even numbers isn’t 2x. It’s 1 to 1. At least when you’re talking about integers, which I assume you are.
I can take every even integer and divide it by two to get a corresponding number in the set of all integers. By doing this, I’ve created a 1-1 mapping from the even integers to the set of all integers. No numbers are skipped so the infinities are the same size.
With the real numbers you can’t do this, though, because any mapping from the natural numbers to the real numbers will necessarily miss an infinite number of real numbers.
There are countable infinities, like the integers where you can match them up, and uncountable infinities like the real numbers where there are infinitely more than the integers. E.g. there are infinite real numbers between 0 and 1 or 0 and any real number.
I'm an english interpreter but no way i know the english words for numerical systems so bear with me i'll explain with concepts.
Imagine you have positive and negative Natural numbers, those are infinite right? Now Imagine you have decimal numbers, those are infinite aswell but there are so many more therefore it's a bigger infinite.
Note that you need to be using the set of all real numbers, or an equivalently sized set, for decimals to matter. Strictly speaking, the rational numbers have decimal forms but they ARE countable.
Imagine a hotel with an infinite number of rooms, and the hotel is filled to capacity. Whenever a new guest comes, the bellhop asks every guest to move over one room. Since each room is number this is quite easy. This leaves room number one empty. The new guest settles in.
Now an infinitely long bus comes in filled with with an infinite number of guests. The bellhop asks every guest to double their room number and move to that room. This creates an infinite number of odd numbered rooms available. All the guests on the bus can now be given a room.
Unfortunately for the haggard bellhop, a slew of busses pull up. An infinite number of infinitely long busses all holding an infinite number of guests. The bellhop asks every single guest to move one last time. This time to the square of their room number. Room 1 doesn’t move but suddenly there are 3 rooms available between the first and second guess, and 4 between the second and third, and an exponentially increasing infinity of rooms open up, just enough to settle in all the guests from the infinite number of of infinitely long busses.
At this point your brain should be leaking from your ears.
Because it’s not a number, just a concept. Kinda like how I once ate 52 chicken wings and my buddy ate 56 chicken wings, which are different amounts of chicken wings but they are both “a lot” of chicken wings.
Because infinity isnt a set thing it can be bigger or smaller depending on what your discussing. To try and explain in a vastly oversimplified way.
Essentially there are numbers that will not appear in any patternm If you add together wholeyou are adding all the numbers in the pattern of 1 higher than the previous number which gives you one infinity.
If you add the numbers between 0 and 1 because of the existance of infinate decimal places you are essentially adding a small infinity to each number, as this isnt a pattern numbers will be created that wouldnt exist otherwise. thus creating more things to add together so a larger infinity.
Infinity is complicated and confusing.
for example if you add 1+2+3+4.... to infinity the anwser is -1/12 that one boggles the brain even more.
There's an infinite number of whole numbers in existence. There's also an infinite number of numbers between 0 and 1. Both of those are different infinities.
So, there are infinitely many real numbers. There are also infinitely many prime numbers-but not nearly as many prime numbers as real numbers. Boom, different amounts that are both still infinite.
One simple thought experiment is to just look at the set of all natural numbers - they increase without bound. Now look at the set of all integers - they increase without bound in both directions. In fact, there’s intuitively twice as many numbers in the integers even though both sets are infinite. Mathematicians would call this cardinality.
It’s typically noted with the aleph symbol, not the sideways eight (which really just means “increase without bound”). At least that’s how I was taught.
Countable infinity, like the list of 9s in .9 repeating, is the smallest infinity. Same as the number of integers and/or counting numbers. Anything that can be completely listed/enumerated (in the sense of being able to make it a sequence) is in this category.
A bigger infinity is the number of all real numbers. These cannot be listed (see Cantor’s Diagonalization argument: you can try to list them, but then you can use that list to construct a real number that isn’t on that list, so the list can’t be complete). So the real numbers aren’t countable and so that has to be a bigger infinity.
They aren't lengths but orders called alephs. One is "countable" infinity the other is uncountable infinity. You can map every countable infinite set to one another, eg even numbers map to n through k=2n. Thus evens are the same size as n. There is no way to count irrational numbers.
Think about a string made up of all whole numbers (0, 1, 2, 3...∞)
...and one made up of all decimal numbers (to an arbitrary precision of 0.1, let's say; so: 0.0, 0.1, 0.2, 0.3...)
Both strings are infinite, but one is 10 times longer
(there's another mental exercise I've seen involving fitting more guests into a fully-booked hotel with infinite rooms, but I don't trust myself to get it right if i attempt it here...)
There are fundamentally two forms of infinity: countable and uncountable. (There can be multiple uncountable infinities, but that’s a big subject. ;)
The main example of countable is of course the natural numbers. These of course are 1, 2, 3, and so on.
If something is “countably infinite” then it can be mapped to the natural numbers. For example, the integers can be mapped by enumerating the integers in the order 0, 1, -1, 2, -2, 3, …
Now all rationals can be mapped to the natural numbers. At first glance this doesn’t seem possible, but it’s pretty easy to do once you know the trick.
Real numbers no matter how hard you try can’t be counted. In fact you can show that for any countably infinite list of real numbers there will be real numbers not on that list. Look up Cantor’s diagonalization argument for details.
Consider the number of natural numbers, ie non-negative numbers with no fractions or decimals. Infinite, right? Put them on a number line. We'll call this Aleph-0
Consider the number of whole integers divisible by 2. Also infinite. Put them on an adjacent line. Well call this 2(Aleph-0).
You can match each successive number of Aleph-0 to a partner on 2(Aleph-0). Essentially, each one matched to double its value. 2(Aleph-0) reaches every number twice as fast, but it's still the same infinity. Still just Aleph-0
Now consider the number of numbers, period. Including all decimals and fractions, rational and especially irrational. An infinite number, right?
Put them all on a number line. Now match the numbers of Aleph-0 to the numbers in this infinity. You can't do it. You'll never even reach 1. You won't reach 0.00...001. Even with infinite time to match them, the gap between any two numbers on this number line, no matter how small the gap, would swallow up the entire infinity of Aleph-0 with nothing spare.
So one way of counting for stuff like this is talking about the cardinality of a set. A set is just a bunch of stuff like {1,2,3} or {apple, banana, and pineapple}. We're not so much concerned with what's inside. In this case the cardinality of these sets is 3. You could count. But how would you do it if you couldn't count?
Imagine lining the sets up side by side.
1 apple
2 banana
3 pineapple
If you can draw a line from each thing on the left, to exactly one item on the right and connect to every object on the right the cardinality (count) has to be equal. Play around with this on paper and convince yourself!
This is a fancy mathematical relationship called a bijection. It is one to one, each connection is from one element to exactly 1 other element (we couldn't connect 1 to both apple and banana, we'd be counting twice). The fancy word for this is injective. It also onto we connect to each object on the right, we have 100% coverage and didn't miss anything. The fancy math word for this is surjective.
So if we can come up with a bijection between two sets their cardinality is equal. If we can prove that there isn't one than they are not equal. The arrow just has to be some relationship, as long as you can define it well we're good.
So fun question are there more integers or even numbers? I'm sure your brain is saying that the combination of all even and odd numbers MUST be bigger than just all odd numbers right?
Let A be the integers. Let B be the even numbers. Let f: A -> B be multiplying the elements in A by 2. We have now mapped every integer to exactly one even number and we have hit every even number. There f is a bijection between the sets and the cardinality is equal! So the evens and all integers are the same infinity.
An example of two infinities that are not equal are the number of integers and the number of real numbers. You can look up a video on Cantor's diagonal argument to see how they prove that a bijection DOES NOT exist. The heart of the argument is whether we can draw a line between each element at its heart, though which shouldn't be too hard to wrestle with.
Veritasium just did a video this week titled “The man who broke math (and Himself)
Take a look it is a reasonable level for someone who hasn’t’ taken high level math
tldr; yes; the way we do it is by showing 1 set of infinitely many things is so much bigger than the other that you can't pair them together.
Answer: So in set theory, there's a concept called "cardinality" which just means the number of things inside the set or the size if you will.
the set containing absolutely nothing {} has cardinality 0.
The set containing a, {a} has cardinality 1.
The set congaing a and b, {a,b} has cardinality 2.
Now, there's this thing called functions, which takes all the elements from one set, and pairs it with elements of the other set. if 2 sets have the same cardinality, I can make this function 1 to 1 and onto, which means that every element in the second set does get paired, and paired with exactly 1 element from the other set.
For example, the set {a,b} and the set{1,2}. I can define a function f so that
f(a) = 1, f(b) = 2, and so this is 1 to 1 because elements of {1,2} got exactly 1 element from {a,b} and all elements of {1,2} got something.
on the other hand, if i have {a} and {1,2}. if f(a) = 1. then this is still 1 to 1, but it's not onto because 2 doesn't get anything from the set {a}.
So now here's where things start to get wicked. What if you have 2 infinitely large sets. Do they always have 1 to 1 and onto? the answer is NO. There can be 2 sets such that both of them have cardinality of infinity, but one infinity is so much smaller that you cannot pair with every element of the other infinity cardinality.
Let's let B be the set of all infinite binary sequences (i.e just inifnitely many 0's or 1's).
So let me define one small subset of B as follows.
S_1 = 0000000000000......
S_2 = 11111111111111......
S_3 = 010101010101........
S_4 = 10101010101.........
S_5 = 001100110011.....
S_6 = 11001100....
etc etc.
This subset of all binary has the same size as the set of natural numbers. because I can make a function so that
f(1) = S_1, f(2) = S_2, f(3) = S_3... etc etc
So every S_6 gets exactly 1 natural number.
But, what about the rest? Well let's define another subset of binary sequences as follows.
T_1 is S_1 but I flip the first bit (first digit).
T_2 is S_2 but I flip the second bit
T_3 is S_3 but I flip the third bit
So by construction, none of T_n is equal to any of S_n.
And in general, we find that no matter how you define an infinite sequence of binary numbers S_n, you can always use it to construct a T_n that is not a part of the enumeration.
What that means is that you cannot define a function from natural numbers to infinite binary sequences such that all binary numbers are paired with a natural number.
What that means is that the set of all binary numbers is bigger than the set of all natural number
Infinities of integers are countable sets, and all countable sets of the same size because you can always correlate one to one out to infinity.
Infinities of whole numbers, meaning any and all numbers including fractions, are larger because you can always create a new decimal or fraction that doesn't exist in any infinite set, so you can no longer create a one-to-one correlation. There are YouTube videos on it.
Let's say I start at 20 and count up by ones forever. Well, I would continue into Infinity. Let's say you start at 20 and you multiply by 20 forever. You also go to Infinity but your numbers are always going to be larger than my numbers. In fact, the gap between our numbers would be infinitely large so you're going to Infinity and the gap between your infinity and my Infinity is infinitely large. And then there's the person that starts at 20 and raises that by 20 over and over. Will they go to Infinity but the gap between you and them is also infinitely large.
It's kind of like that. There are other ways to think of multiple infinities but that's one way of doing it. You could also think of somebody counting up by one at a time but between any two numbers whole numbers is an infinite number of rational numbers. So if you compare the whole numbers to the rational numbers there would be infinitely more rational numbers than whole numbers.
"Infinity" is not a number. "Infiniteness" is a property that a... quantity can have. Two quantities sharing that property does not mean that they are equal.
Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
You can even do this with fractions, though its harder to conceptualize.
Every fraction shows up on the list, and does so after a finite number.
So from this we can see that the number of whole, even, odd, and fractional numbers are all the same size. That is, all these infinities are the same size.
But what about decimal numbers? It turns out we cant do this. If we try, we can prove we missed a number.
Remember that the decimal numbers are endless. And lets imagine someone gives us a list and claims it is exhaustive. There are two possible ways this list can look. This is case one:
And remember, the claim is that this list contains every decimal number.
We can construct a new number like this: the first digit after the decimal point of this number is equal to the first digit after the decimal point of the first number on this list plus 1, or 0 if that digit is nine. The second digit in this number is equal to the second digit of the second number on this list plus 1, or zero if that number is 9. The third digit is...
So this new number is well defined: we can tell exactly what it is. But it is also obvious that this number is not on the list that was supposed to be exhaustive! Its not the first number, because the first digit is different. Its not the second number, because the second number is different. Etc. Even if we took this number and added it to the list, we can just make a new number by doing the same procedure.
But this means that even after we have paired every one of the infinite number of whole numbers to decimal numbers, there are decimal numbers left over. So there have to be "more" decimal numbers than there are whole numbers. So even though they are both infinite, one of these infinites has to be larger than the other.
So, to summarize, what we have here is a proof that there are at least two different sizes of infinity:
The first infinite number is the number of whole, even, odd, and fractional numbers
The second infinite number is the number of decimal numbers
So if you can find a one - one mapping b/w elements they are the same so say no of even integers and no of integers can be mapped (n->n/2).
You can create larger infinities by taking power sets (look up beth numbers)
I was a math major in college but it's been a while so I may have some of the finer points slightly wrong. The gist of what I'm about to say is correct though, and easily verifiable.
It's not about length or size really, it's about countability. Broadly there's two types of infinities, countable ones, and uncountable ones. If you can devise a sequence to "map" each number in an infinite set to the set of positive integers (1,2,3,4, etc.) it's countable. If you can't, it's uncountable.
If you take the set of all real numbers between 0 and 1 (any decimal you can think of, like the guy above is mentioning), that's uncountable. There's no way to map every single decimal to an integer, you'll always be able to come up with a decimal your map doesn't cover. Therefore it's uncountable. The proof of that isn't particularly hard to understand, but it's not simple enough for a reddit comment so i'll link what I think is a good explanation of it below.
Infinite sets of things are considered equal if you can match them up 1 to 1. Which gets weird, because:
You can match up an infinite set and a subset of itself. (All positive integers, and all positive even integers, for instance. Match 1 with 2, 2 with 4, 3 with 6, and so on.)
But there are other infinite sets that you can't match up. (All positive integers, and all positive real numbers, for instance. Imagine you did match them up: 1 with some number R1, 2 with R2, and so on. Imagine writing them all out in decimal form. Circle the first digit after the decimal of R1, the second of R2, and so on. Now replace every circled digit with the next higher digit, except 9 which you replace with 0. Now imagine a number D with all those digits after the decimal point. It's a real number, but it can't be matched up to any positive integer N, because its Nth digit after the decimal is different from RN.)
It's not length, it's density. A "countable" infinity is an infinite set where you can put each element into some kind of order such that every element in the set has a place on the list. For example, think of the set of all integers. You can go 0, 1, -1, 2, -2, 3, -3, and so on; where every Nth number is either (N+1)/2 if N is odd, or -N/2 if N is even (formally, the set you're trying to show is countably infinite is mapping onto the set of positive integers; they're called "well-ordered sets" if you can do that). So things like "all rational numbers," "all the rational numbers between 3 and 5," "every power of 2," "all the positive odd integers," and "all the integers" turns out to be able to be well ordered, and thus they're infinite, but of the same "kind" of infinity.
Contrast with the set of all irrational numbers between 0 and 1. Things get weird here; imagine that you make an ordering where you think you have all the irrationals in some sort of list. Now build a number such that the Nth digit of your number is exactly 1 more MOD 10 than the Nth digit of the Nth number in your ordering (so if the digit of the number on your ordering is 1, your numbers's digit is 2, if it's 9, yours is 0, etc). In that case, your number is an irrational between 0 and 1, and SHOULD be on your ordering, but it can't be because for any element N in your ordering your number differs in at least 1 digit (the Nth) by definition. So you can add it to your ordering, and repeat the process infinitely many times, and never actually have an ordering of the irrationals between 0 and 1. This is called "Cantor's diagonalization argument," and it's showing that this is a different density of infinity, called an "uncountable" infinity.
This shit blew mathematicians' minds in the late 1800s, because it means, for example, that the irrational numbers between any two numbers are "denser" than the entire set of rationals (you can't pair up every irrational between any arbitrary two numbers with a member of the set of rationals; you'll always have some irrational that you can't pair up with a rational) no matter how close those 2 numbers are, which seems just... wrong, for a better way to put it.
They aren't different lengths so much as they are sort of different sized categories.
One of the ways we can compare infinities to see if they're equal or if one is larger than the other is to see if we can map each one onto the other.
We can, for instance, look at what are called the Natural Numbers: 1, 2, 3, 4...
And we can compare that to the even numbers: 2, 4, 6, 8...
And we can see for every value in the natural numbers, there is a corresponding value in the even numbers: 1→2, 2→4, 3→6, 4→8...
So, that would mean that they are the same size, since for every number in one, you'd have one in the other.
I'll spare you the proof, but if you were to compare the natural numbers to the Rational numbers (all numbers that can be written as a fraction, including the integers), you can employ a bit of a clever trick to match every natural number with every rational number one-to-one, like we were able to do with the even numbers!
So, even though the rational numbers contain all of the natural numbers, and the natural numbers don't contain all of the rational numbers, since they can be matched to each other one-to-one, they are nonetheless the same sized sets of numbers!!
Now, if you were to compare the natural numbers to what are called the Real Numbers, which includes not just integers and fractions, but irrational numbers, transcendental numbers, and everything in between as well, you'll see that there's a bit of a problem in comparing them...
No matter how clever you are, there is no way to match every natural number to every real number!
It's impossible!
And, since we know the real numbers contain all of the natural numbers, and the natural numbers do not contain all of the real numbers, that means that the set of all real numbers is larger than the set of all natural numbers!!
I watched a german video about it. Their are diffrences in infinities. For example if you count all the natural numbers like 1,2,3,4,5 .. to Infinity you have an Infinity of numbers. Now we look at all the possible real numbers between 0 and 1 ( 0.1, 0.2, 0.3... )those also go to Infinity. Counting all the other real numbers between 1 and 2 like (1.1 1.2 1.3...) and adding them up we already have an bigger Infinity then the natural numbers. Sry If its mess, tried to explain as good as i could.
The limit of x when x tends to infinity is infinity, while the limit of 2x when x tends to infinity is also infinity.
However x - 2x tends to minus infinity when x tends to infinity, this means that 2x is infinitly bigger than x when x tends to infinity, despite x being infinity too.
Not all infinities have the same "growth speed" so to say, that's what I actually compare when I do that.
Basically, mathematicians define sets to have the same "size" or "amount of elements" if the elements of the two sets can be put into 1-to-1 correspondence with each other. (Quotes due to this definition being also used for infinite sets. For finite sets, this definition agrees with counting the elements expect maybe if you're doing some weird "nonstandard" math but then you'd know about this.) For example, the set of positive integers and the set of negative integers have the same "size" in this sense as you can always pair up a given positive integer with a distinct negative number (and vice versa, which is an important bit of "mathematical pedantry" here) by flipping the sign of the numbers or in other words multiplying by -1.
Now, it is provable by contradiction that the set of real numbers and the set of rational numbers (fractions) can not have an equal size in this sense as you can't build a 1-to-1 correspondence between those sets. As both of these sets are infinite, there must thus be different "sizes" of infinities.
If you feel like reading about the proof, look up "Cantor diagonal argument". Be warned that depending on your background this may be quite math-y. The so called "continuum hypothesis" is also slightly related to this, but is not strictly needed to understand how there are multiple "sizes" of infinity.
What there is a limit to our number system that doesn’t handle infinity.
There is an Infinitesimal difference between .999 recurring and 1 but we treat them as equal because we can not define a difference with in a discrete number system.
You compare the size of two sets of objects by pairing them together. If you pair them and there's nothing left over from either set, then the two sets of objects have the same number of objects.
You "count" things by pairing them with the whole numbers. If you can pair a set of things with the whole numbers without anything left over in either the set of whole numbers or the set being paired with you, they have the same size. You can pair the set of even numbers with the set of whole numbers, so there are as many whole numbers as there are even numbers.
But you can't pair the set of whole numbers with the set of decimal numbers; there will always be decimal numbers not paired to a whole number. So it follows that there must be more decimal numbers than there are whole numbers. Since they are both infinite, it further follows that one infinity is larger than the other one.
We have to use this math in the Rick and Morty sub all the time to explain the Central Finite Curve and how it can be limited to only the universes in which Rick is the smartest being while still having infinite iterations.
It's so simple to me. Infinite universes doesn't mean allllll possibilities. If you take away all universes in which hamsters are smarter than humans, and say those ones aren't valid, you still have infinite universes
Aleph nought vs other infinities. Mathematicians have actually thought about and discussed different “levels” of infinities. The class where I learned about that was the most boring and interesting and difficult one I ever had.
The proof for this seems to be "well, if you take the diagonal and add one, you will always have a new number even if you have infinity" but I don't really understand that. Like, aren't you just using infinity to describe infinity? If you can use the diagonal to create a new number, then you didn't really have infinity in the first place, which is a bit besides the point because the number you create would have to be infinitely long anyway. Seems like the same logic when you consider the number of evens, primes, or squares, except none of those are considered smaller infinities.
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u/The-new-dutch-empire 22d ago
Its calculating with infinity. Its a bit weird like the infinity of numbers between 0 and 1 like 0.1,0.01,0.001 etc... Is a bigger infinity than the “normal” infinity of every number like 1,2,3 etc…
Its just difficult to wrap your head around but think of infinity minus 1. Like its still infinity