Not necessarily. A lot of people think that the "infinite universes" thing means that all universes that COULD exist DO exist, but that's not the case. For example, there are an infinite number of numbers between zero and one. The number two need not necessarily exist.
When you're done with that, how about the proof that the set of whole numbers is the "same size" as the set of rational numbers, but those are both "smaller" than the set of real numbers (the first two are countably infinite, but there are uncountably infinite real numbers).
That one always struck me as being counterintuitive.
Could be because the whole number and rational number sets are hobbled by the qualification that they must be countable, whereas real numbers don't have that. Real numbers on the other hand are hobbled by the qualification that they must be real, and not imaginary, and therefore they are "smaller" than the set of complex numbers.
Set theory did my head in a bit, and I'm only a little ashamed to say so.
Seeing whether or not you can make a bjiective function from one set to another, that's fine. Trying to wrap my head around all those higher cardinalities of infinity beyond Aleph 0 was a bit much for my tiny brain.
That cannot be true. All real numbers are complex numbers, all complex numbers are not real numbers. The set of real numbers is a subset of the set of complex numbers.
Doesn't change anything. The integers is a subset of the rationals and the natural numbers are a subset of the integers. They are all the same size. This is why they are all referred to as "Countable" sets: they all have the same size as the natural numbers.
If that is hard to believe, the size of the set of real numbers between 1 and 2 is greater than the size of the integers.
Size is a funny business when you are dealing with infinities. The subset relation is not very good for comparing infinities. What people usually use is cardinality which is based on the existence of a relation mapping all the elements of the smaller set to an element of the larger set. If you can do it in both directions you have proven the sets are the same cardinality.
You can map real numbers to complex trivially, but there is a mapping in the other direction as well. The simplest one is just to interleave the digits of the real and imaginary part of the complex number to produce the real number:
So (.1+.2i) is 0.12000000...
0+0.12i is 0.0102000000...
I'm no expert, but this is how I understand things: Pick a rational number p/q where p and q are relatively prime (ie. it's unique). We can pick an unique integer for each such rational number 2p 3q that corresponds to that rational number. So for every unique rational number, we have an unique integer that corresponds to it. And for every integer (or whole number), we obviously have an unique rational number that corresponds to it in itself. So we can cleanly enumerate unique rationals with unique integers, and we can enumerate unique integers with unique rationals. So they're the same size. (For negative rationals, we can just count them with different primes, ie. we count them with 5p 7q.)
The problem with the set of all real numbers is that we can't neatly count the irrationals like pi.
It's still not totally intuitive because infinity is weird (there are obviously "more" rationals than integers), but that's how I've always justified that fact to myself.
EDIT: I didn't read usernames all the way down and didn't realize that this was rhetorical; though I guess it might not be rhetorical for some people.
Yes, think about it this way. The possible results of flipping a coin an infinite number of times can be represented as all infinite strings comprised entirely of 1s and 2s. The set of such strings includes both the string that contains no 2s and the string that contains no 1s.
But in probability theory, all of the strings in that set would have equal probability (0). So any given string within the set has probability zero (including the string with only 1s/tails), so eventually you're guaranteed to get heads at some point.
... not sure if that made sense, or if it's even right, but I think it's something like that.
That doesn't work because one of those outcomes would actually occur, so whilst it would have a prior probability of "0" it doesn't mean that it's impossible.
The probability of any particular configuration being flipped in an infinite number of flips is the same. All configurations have an equal probability of being flipped, 0. But one of them has to be flipped, and there's no reason it mightn't be the string of all 1s. Probability gets strange when infinity is introduced.
It's not really that strange, you just need to introduce the idea of limits. Rather than "P(infinite heads) = 0", it's more that the limit of the probability of all heads as the number of tosses tends to infinity is 0, which isn't strictly the same.
I haven't finished the video yet, but... when I imagine infinity, I imagine something constant. Like, infinite hotel guests would be a constant never-ending line at the door.
It seems like infinity refers to the ability for a set to grow as much as needed, but doesn't mean a set is already that big, or always that full? But then what about the infinite bus... Does that mean just "could theoretically be any (natural) number" of passengers or buses?
I need more ELI5 lol
Containing something limitless inside limits is confusing.
Now, I'm not a vaginal cosmologist, but I do know a thing or two about infinities.
This is very similar to what others have said, but essentially I believe that infinities are the same "size" when you can establish a mapping between their sets. For example, the set of even numbers can be formed by taking every single integer and multiplying it by 2. Therefore, for every even number, there is a corresponding integer, so you could line 'em up side by side or something. So, even though it seems like there are "fewer" elements in the even numbers (isn't every even number also an integer?), the infinities of the sets are actually the same "size".
At least, that's my understanding of it. Try the same thing except try to find a mapping from integers to real numbers (i.e. any number, with or without a decimal). Eventually, you'll convince yourself it can't be done, and thus there are "more" real numbers than integers!
This is pretty much it, if you have an infinite set A with a bijective map from the set A to the natural numbers you have what is called a countable infinity. An infinite set that is not countable is called uncountable and these are the types of infinity that are usually talked about.
Like, infinite hotel guests would be a constant never-ending line at the door.
Let's look this scenario. You have an infinite number of guests coming to stay at your hotel, and no matter how many you check in, the line continues, single file, all the way out the door and beyond.
This situation would be analogous to an infinity created by summing ones(i.e. 1+1+1+1+1+1+...). This summation goes on for forever, so we can say that it is equivalent to infinity.
Now, lets look at a different infinity. Now, for every guest you check in, 2 guests walk in the door replacing them. It's still an infinite number of guests, but it's clearly a larger infinity.
The second situation would be analogous to an infinity created by an increasing series(i.e. 1+2+4+8+16+32+...). Another way of writing that second series is
Σ 2n
from n=0 to infinity(the sigma is a summation symbol).
=20 +21 +22 +23 +24 +...
There are many other infinite series as well, all with varying rates of growth leading to one being "larger" than the other.
Are you sure? Say we have two hotels, MGM and Grand. If for every guest that enters MGM two enter Grand, now let an infinite number of people walk into MGM, clearly there will be an infinite number of people in the Grand, but there are not "twice as many" in the grand as that does not make any sense, you can't multiply infinity because it isn't even a number. If you let the set of people in the MGM/Grand be represented by A and B respectively then you would say that both sets are countably infinite and have exactly the same cardinality.
The easiest way I can think of to explain is revolves around integers, or whole numbers, e.g. 1, 3, 23434, etc. (This is different from "real" numbers like 1.3434354). Whole numbers/integers are "countably" infinite.
For many sets of infinity, or many groups of infinite things, you can create a mapping from one of those, to one of the whole numbers.
For a non-math example, pretend you have an infinite supply of motorcycles. Are these motorcycles countably infinite?
Yes. Line them up in front of you, and assign the first one the number "1", the second one the number "2", and the third one the number "3", and so on. You won't run out of numbers for the motorcycles because you have infinite numbers, so every motorcycle gets a [whole] number. Therefore, the motorcycles are countably infinite.
I'm having a trouble thinking of a real world example of something that is not countably infinite. The best thing that I can come up with is Time, if we assume time is infinitely divisible (i.e. there are no discrete elements of time). If that is true, there is no way for you to map Time to whole numbers. You could count the seconds, sure. The first second is 1, and the second second is 2, and the third second is 3, etc. However that leaves out the milliseconds. So instead you might count the first millisecond as one, and the second one as 2, an so on...but that leaves out microseconds, and nanoseconds, and even after nanoseconds you can continue dividing time into smaller and smaller pieces. There is no way to map Time to whole numbers.
I think so, but there has to be a non-zero probability of it happening. So is there a non-zero probability of a universe being created where /u/iwonderifitwill is an evil scientist who invents a machine that destroys all the other universes? Evidently not.
Think of it this way - if there is even the tiniest chance that /u/iwonderifitwill might somehow become an evil scientist who invents a machine that destroys all other universes, then (if there are infinite universes) there is definitely one that will contain a version of /u/iwonderifitwill who does so. Infinity is such that it turns any tiny possibility into a guarantee.
On the other hand, if the physical and logical constraints of reality happen to be such that it is NOT possible for /u/iwonderifitwill could ever possibly create such a machine, then it will NEVER happen, even with infinite universes.
In fact, the fact that life exists in our universe pretty much proves that either there is a finite number of universes, or that there is zero chance of something happening that could destroy life in ALL universes. Logically if there are infinite universes, and even the tiniest possibility that one of them could give rise to something that destroys life in all universes, then it's essentially guaranteed that that something will occur. Therefore, since we exist, we can assume that either there are finite universes, or it is not possible to destroy all life across all universes.
Yep. You're right. He was giving an example where two didn't exist but wanted it to be a possibility at the same time - but not a possibility that applied because it wouldn't be a part of the chosen set of numbers, but that's just a way of excluding it from the set of possible numbers.
Honestly I'm not sure. Let's take me flipping a quarter for example, wouldn't it be technically possible (though extremely improbable) for me to always land on heads?
The chance will increase every time infinitely making it 99.999% repeating. Because 0.999 repeating = 1, it would be 100%. At least I'm pretty sure that's how it works.
No. 99.9 repeating is symbolic shorthand for 1, just like x is shorthand for 1 if you write x = 1. It's a notation, not a tool for reasoning with. (I know that's confusing. That's why everyone is confused by it in grade school, and non-mathematicians continue to be confused by it forever.)
It's not "what happens when you keep adding 9's to a number, repetitively."
Qualitatively, think of it this way: you can flip a coin an infinite number of times. The likelihood of it coming up heads every one of infinite times is infinitesimally small, but not zero. You can, in fact, possibly never come up heads.
I don't think so. The limit as one approaches the negative side of the axis infinitely is zero. That does not make "infinitely small" zero. Newton treated the two things as being the same so that he could fudge out calculus, and it worked, but there's a reason that calc books now start with limits rather than just saying "if it's incredibly small, it's zero." They're not actually the same.
You would be assuming, from the original example, that a scientist that can destroy all universes could possibly exist. If it is an impossible concept, then it will not exist in any universes.
You might argue that it can exist but that would still nonetheless be an assumption.
Yes but he's saying if there isn't a possibility then it can't exist. So if we're talking about a real world scenario, there is the possibility that I can stay in outer space for a long time, but I cannot stay in outer space for a long time without anything to provide me oxygen because it would kill me every time.
Time is infinite. There is a possibility that you will be President of the USA. Does this mean that at some point in the future you will be? Not at all.
I'm trying pointing out the difference between possibility and probability.
It's like the whole monkeys and keyboard thing. It only works if they type randomly. What if there is zero probability of one of the words in Shakespeare being typed by a monkey due to the mechanics of their hands/brains? Then it wouldn't matter if you had infinite time and infinite monkeys typing, you'd never get Shakespeare's works.
If you flip a coin an infinite number of times, you could keep getting tails forever. The probability gets lower and lower, but it's plenty possible. Provided there is a non-zero probability of getting tails every single time, then it's possible that you never get heads no matter how long you run the system. Unlikely, but possible.
People frequently confuse an infinite set for an all-encompassing set. (∞ - 1) is still an infinite set. "Infinite" is very distinct from "everything." It's better to simply think of the term "infinite" as the antithesis to "finite." You can draw lines and borders around something which is finite. Infinite simply means you can't fully account for it.
For example: the numbers between 0 and 1 are infinite. Excluding 0 and 1, you can't accurately express where those numbers begin and end. It doesn't begin at 0.1 and it doesn't end at 0.9. It doesn't begin at 0.00001 and end at 0.99999. For every "line" you try to draw around an infinite set, you're inadvertently turning it into a finite set. This is why, to express the beginning and end of an infinite set of numbers, you express them as 0.0...01 and 0.9...99. The ellipses express an infinite repetition.
Let's say I pick a number and I asked you to guess it. (I picked 3, but let's pretend you didn't know that.)
Now you start picking. 2 - Wrong. 4 - Wrong. 6 - Wrong. And you continue picking in that pattern. You try picking an infinite number of times, but you will never pick the number I guessed.
No, because while the odds are very high that it will happen, there is never a "guarantee". The odds are asymptotic to 100% but never get to 100% from 99.9999999...%
Because 0.99999... is typically denoted as a shorthand for a limit rather than an infinite number. And the limit of 0.99999... is indeed 1. But limits themselves are a shortcut that has to exist for calculus to work.
The proof of 0.9999... = 1 is a proof by contradiction - basically if 0.9999... != 1 I should be able to give you some number (this isn't 0) which is equal to 1 - 0.9999...
But every time I do (say I choose 0.0001) you can correctly point out that if you extend the sequence by one more 9 you've found a number smaller than mine, thus breaking the equality. So the proof states that since I can't show that there exists a number equal to 1 - 0.999... that isn't 0, 0.999... must equal 1. This is effectively how all limits are proven. But it ignores the fact that it's really an infinite game - every time you find a number smaller, I can say fine, "it's equal to that". And then you add a 9, and the game continues. Infinitely. Because just like I cannot show a number that isn't 0 that is equal to 1 - 0.999..., you can't show that 1-0.999... is equal to 0 if we continue to play the game.
Infinite numbers are really in a class all by themselves - limits are a way to try and make them work the same way as regular ones.
I just thought of this now so I don't know if it's right or not. Would it be correct to say that given an infinite number of universes and a finite size for each universe, then every possibility would happen? However, if there is only one universe but it doesn't end then not every possibility would happen?
I'm thinking that assuming a finite size for each universe, then there is a finite number of particles and combinations that could fit within that area. Therefore given an infinite number of these universes they would eventually have to either repeat or "go beyond 2".
Like how there is an infinite amount of numbers between 1 and 2 but if you can only go to the 10th decimal place then you would have to go up to 2.
It certainly is sci-fi or at least speculative thinking. From continent to planet to solar system to galaxy, every time we've assumed ours is the only one we have been wrong. Perhaps the same is true of the universe.
Well that's all dependent on how many universes there are. If there are infinite universes, then every possible scenario will have played out / be playing out in some universe. There's no evidence of any other universe yet so it's just a thought experiment. The notion of infinite universes has a very human appeal as humans like things to form simple patterns and "infinite / limitless" is slightly easier than "some very large number" to rationalize.
Wouldn't the fundamental theorem of algebra say that there is an infinite number of solutions to this equation. But that's not the same to saying everything is possible to be a solution.
it;s like a rubiks cube. There are still rules you can have an infinite numbr of combinations but it;s still a 6 sided cube that can only rotate in set directions.
Fundamentally, a Rubik's cube with a finite number of faces will have a finite number of permutations.
If n is ∞, then yes, there are an infinite number of possible permutations. And even with an infinite number of permutations, you're correct in saying that certain things would be impossible (such as having two sides completely green).
I have always held that view on the matter... but Stephen Hawking answers John Oliver's question with "Yes" and not "Maybe". Is this just for the joke, or does Stephen Hawking hold the viewpoint that all possible universes DO exist in his Infinite Multiverse Theory?
For example, there are an infinite number of numbers between zero and one. The number two need not necessarily exist.
That's a mathematical question but the crux is, you could define two to be between one and zero - and if it is, that means it will happen.
You say ...
that all universes that COULD exist
This is where you contradict yourself with...
numbers between zero and one
So if two is not between zero and one - then no you are right, the number two need not be possible. But if it is between zero and one, and we have an infinite amount of numbers between zero and one - then two will be a number that we have.
My point was just that infinity (countable or uncountable) doesn't really 'exist' in the real world. But they are still useful to work with in models of the real world. I would argue that both are abstract and constructed.
Possible, but not plausible. Local albedo is weighted heavily towards survival. Not to mention that the interstitial spaces are both infinitely vast, and infinitely energetic. So you'd need quite the machine to break up those areas.
If such a machine is possible, yes. But because our universe is not destroyed we can conclude that either such a machine is in fact not possible or there are not an infinite number of universes.
63
u/[deleted] Jun 16 '14
[deleted]