Comments by "eggynack" (@eggynack) on "TED-Ed"
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An infinite hotel may not have rooms available if you have infinite people. Take all the hotel's guests and line them up, assigning each a number equal to their place in line. Put person one in room one, person two in room two, and so on. Now, for every room, there is a person. Or, equivalently, it's impossible to name a room for which there is no person. Thus, the hotel is fully occupied. It's also possible to assign people such that any number of rooms, from one through infinitely many, are unfilled, or such that every person gets infinite rooms, or such that every room gets infinite people. The possibilities are literally endless.
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Countably infinite is quite possible. It just doesn't mean what you think it does. A set is countably infinite if you can create some list such that, for any number in the set you name, you can count to it going by that list. So, for the natural numbers for example, the list would just be 1, 2, 3, 4, 5... If you name any natural number you will eventually reach it just by counting upwards like that. Thus, the natural numbers are a countably infinite set. The list need not be a straightforward ascending order one though. The rational numbers are countable, for example, in spite of the fact that a straight up ascending or descending list would be completely impossible.
More broadly, this video's presentation of infinity is an accurate one, and infinity is a reasonably comprehended and understood thing in mathematics. If it weren't then we wouldn't even have stuff like calculus. Really, infinity shows up all the time in math, in basically any field you'd care to study. It doesn't constitute any sort of real barrier either.
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The idea of there being infinite numbers is not assumed to any extent. It's pretty trivially provable. After all, for any finite quantity, there is always a greater one. Numbers don't cease to exist when we stop using them. If I only count to 999,999, then I haven't just named the highest number, until I later count to a million when that becomes the highest number.
And, no, we are not "creating" a number by moving someone. Assign each hotel room a natural number, and put someone in it. Then, move everyone up a room. What room number is being created? The first room obviously existed from the beginning, so it would necessarily be the "last" room. However, because there are infinite rooms, there is no last room, either before or after the movement.
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For any given infinite set, you can ask, is it possible to create a list of the elements in this set such that I can use the list to reach any arbitrary element in finite steps? For the natural numbers, the method of counting is clear. 1, 2, 3, 4, 5... You'll get everywhere just using that list. For, say, the integers, it's a little less obvious. However, you can do something like 0, 1, -1, 2, -2, 3... The same is true for the primes, the rationals, all kindsa sets. These are countably infinite. However, such a task is impossible for the reals. Thus, it is an uncountably infinite set. And, given that these sets are entirely composed of numbers, I'd say numbers decidedly do not become irrelevant with the concept of infinity.
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Y'know, I usually use a mapping for this, but I'ma try something new. Imagine that every room, as part of its design, features a seed at its center. A property of the seeds is that, after 15 minutes, they sprout into guests. So, after that 15 minutes, every room will have exactly one guest in it, and because every room has a guest, the hotel is full.
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N+1 is meaning a lot of different things right now. I assume you're talking about the guest movement strategy here. Yes, at first, for every room n+1, where n is a given guest's starting position, that room starts out full. However, what is possible is moving every guest up a room simultaneously. Guest one ordinarily wouldn't be able to move to room two, but guest two is going to room three. Guest two wouldn't ordinarily be able to go to room three, but guest three has moved to guest four. And so on. This process never stops working, because the hotel is infinite.
The narrative of the word problem is not inconsistent. When the problem says, "The hotel is full," what is meant is that, for every room, there is a person in that room. What is usually meant by that, what is not meant here, is that the hotel cannot accommodate an additional guest.
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Your dad's explanation was inaccurate, I think. I'm not precisely sure what you mean by "all numbers", but the set of positive odd numbers, for example, is not smaller than the set of positive integers, in spite of the fact that it seemingly has fewer elements. To show this is the case, you just have to create a one to one mapping from one set to the other. So, simply map each element of the positive integers to 2n-1 in the odd numbers. Every element of each set will be counted exactly once. There are larger infinities, however, For example, the set of all real numbers between zero and one is larger than either the positive integers or the positive odds.
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Yeah, that's what it sounds like. Basically, any set of numbers that you can put in any kind of list is going to be exactly the same size as any other set with that quality. So the whole numbers and odd numbers are obviously like that. The integers are a bit less obviously like that, because you can list them like 0, 1, -1, 2, -2...
The fractions are not at all obviously like that. For that ordering, you want to make a chart with the integers hanging out on the X and Y axis, and then the entries in the chart use the X axis as the numerator and the Y as the denominator (the Y axis excludes zero). Then you make this big zig zagging line that goes through every entry in the list, remove any repeats, and you have a full ordering of the rationals.
Even less obvious is the set of algebraic numbers. Those are the real numbers that you can write as a finite algebraic expression. So, like the square root of 2 would be an algebraic number. Phi is an algebraic number too. But something like pi cannot be expressed that way, and so is not algebraic. The ordering there is relatively straightforward. You can kinda treat each part of the expression as a digit, and then remove illogical combinations.
The reason all these sets have the same size is because you can easily create a mapping of the sort I mentioned. All the elements are already in a list, so there's a first element, a second element, a third element, and so on. All you need to do is pair the nth element of set A to the nth element of set B and the mapping is done. The real numbers cannot be listed this way, and so that is a bigger set.
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