TREE is an extremely fast-growing function in set theory. TREE(1) equals 1, TREE(2) equals 3, and TREE(3) equals a number so large that its lower bound easily dwarfs Graham’s Number.
Graham’s number is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to Graham’s number.
I am one of them. I still can’t get past the Hotel paradox. To me an infinite number of guests cancels out an infinite number of rooms.
Infinite guests = infinite rooms
Infinity + n = infinity
To say the bus of unbound guests could just move into infinite rooms seems to give a property of rooms without limit that is not shared with the original infinite guests.
The original premize states the hotel is full. Because the only thing that matches infinite rooms are infinite guests.
Apparently I am very stupid. My sister was right all along.
Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
Infinite hotel has infinity guests. You have all the guests move down 10 rooms. Rooms 1-10 are now free. Zero to Infinity and 11 to infinity are equally infinity, since numbers extend into infinity.
In the same manner if you have one set of infinite guests occupy all the even numbered rooms, you will still have an infinite number of rooms open, because the set of all odd (and even) numbers extends infinitely. You could have the first set of infinite guests take each hundredth room (100, 200, 300, etc), then the next set take 99, 199, 299, etc. in that way you could fit 100 sets of infinite guests.
It just illustrates that infinity is not an easily intuitable concept.
What doesn’t make sense to me is infinite rooms and infinite guests and is full. You ask everyone to move down 10 rooms, why is 1-10 now free? You had infinite guests too, wouldn’t more filled rooms appear?
Or Is infinite only infinite (undefined) on the upper end, but defined on the lower? E.g. 1.
You can define the start of an infinite series, just not the end. (Except as ∞ or -∞). You could also have an infinite set that extends both ways.
0 to ∞ contains an infinite amount of numbers. But so does 11 to ∞.
More filled rooms do not “appear”, the rooms just go on without end. These is no “last” guest who moves into some previously unoccupied room. It’s just… endless. Infinite.
It really only makes sense in abstract. Our minds aren’t built to deal with infinity.
There are different “kinds” of infinity. For example, there is an infinite amount of natural numbers, and there is an infinite amount of real numbers. Still, natural numbers only make up a tiny part of real numbers, so while both are infinite, the set of real numbers is bigger. Hilbert’s Hotel is an analogy meant to convey how to deal with these different notions of infinity.
Not really. The guests move to a room with double the number, freeing up an infinite number of rooms.
So the change is from natural numbers to even numbers, freeing up odd numbers. Those infinities are the same, but you can still do this because infinities are weird.
I don’t get it. Is “tree” some kind of notation?
TREE is an extremely fast-growing function in set theory. TREE(1) equals 1, TREE(2) equals 3, and TREE(3) equals a number so large that its lower bound easily dwarfs Graham’s Number.
Yo you don’t gotta throw shade on Grahams number like that, it’s doing its best
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Yes, TREE(3) is larger than a googolplex or even graham’s number.
Why do I always feel like I need a PhD to understand even the first paragraph of Wikipedia articles about math. Is that just me?
yeah where’s the Simple English Wikipedia article for Graham’s number
Graham’s number is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to Graham’s number.
Go get your PhD and write one yourself!
That article is not comprehensible to most people
https://www.youtube.com/watch?v=3P6DWAwwViU
Here is a Numberphile video that describes how large a number we are talking.
Thanks, I get it now!
I am one of them. I still can’t get past the Hotel paradox. To me an infinite number of guests cancels out an infinite number of rooms.
Infinite guests = infinite rooms Infinity + n = infinity To say the bus of unbound guests could just move into infinite rooms seems to give a property of rooms without limit that is not shared with the original infinite guests.
The original premize states the hotel is full. Because the only thing that matches infinite rooms are infinite guests.
Apparently I am very stupid. My sister was right all along.
So, some infinities are bigger than others.
How many numbers are there? An infinite number.
How many even numbers are there? An infinite amount, but half the size of the first infinity.
This is how there are empty rooms in the infinitely large hotel with infinite guests.
Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
Infinite hotel has infinity guests. You have all the guests move down 10 rooms. Rooms 1-10 are now free. Zero to Infinity and 11 to infinity are equally infinity, since numbers extend into infinity.
In the same manner if you have one set of infinite guests occupy all the even numbered rooms, you will still have an infinite number of rooms open, because the set of all odd (and even) numbers extends infinitely. You could have the first set of infinite guests take each hundredth room (100, 200, 300, etc), then the next set take 99, 199, 299, etc. in that way you could fit 100 sets of infinite guests.
It just illustrates that infinity is not an easily intuitable concept.
What doesn’t make sense to me is infinite rooms and infinite guests and is full. You ask everyone to move down 10 rooms, why is 1-10 now free? You had infinite guests too, wouldn’t more filled rooms appear?
Or Is infinite only infinite (undefined) on the upper end, but defined on the lower? E.g. 1.
You can define the start of an infinite series, just not the end. (Except as ∞ or -∞). You could also have an infinite set that extends both ways.
0 to ∞ contains an infinite amount of numbers. But so does 11 to ∞.
More filled rooms do not “appear”, the rooms just go on without end. These is no “last” guest who moves into some previously unoccupied room. It’s just… endless. Infinite.
It really only makes sense in abstract. Our minds aren’t built to deal with infinity.
Thanks. My mistake was to view infinite as strething without end in both directions. Today I learned. Thanks.
There are different “kinds” of infinity. For example, there is an infinite amount of natural numbers, and there is an infinite amount of real numbers. Still, natural numbers only make up a tiny part of real numbers, so while both are infinite, the set of real numbers is bigger. Hilbert’s Hotel is an analogy meant to convey how to deal with these different notions of infinity.
Not really. The guests move to a room with double the number, freeing up an infinite number of rooms.
So the change is from natural numbers to even numbers, freeing up odd numbers. Those infinities are the same, but you can still do this because infinities are weird.
What about our friend TREE(Graham’s Number)
Numberphile did a video on TREE(g64) so you gotta go bigger.
BB(TREE(3)) :)
TREE(Rayo(10100))
TREE(yomomma)
Needs more recursion
Also is your username from stormlight archive?
:)
Justice for phenadora 😭
Haven’t seen them in a while. Hope they’re doing well