Hilbert's paradox of the Grand Hotel
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Hilbert's paradox of the Grand Hotel
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Re: Hilbert's paradox of the Grand Hotel
#2I don't get it, why is this a paradox?
Infinite guests in an infinite never ending carousel of rooms in a hotel, why is this a thought experiment?
It is just logic, without any fucking practical common sense.
Re: Hilbert's paradox of the Grand Hotel
#3>"If an infinite set can be put into one-to-one correspondence with the natural numbers (N) it is called a countable set. Otherwise it is uncountable."[1]
This paradox hinges on the strange notion of cardinality of infinite sets. Specifically, the set of all even integers, the set of all odd integers, and the set of all integers(!) have the same cardinality, and therefore the same "size".
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[1]http://www.math.ups.edu/~bryans/Current/Journal_Spring_1999/...
Re: Hilbert's paradox of the Grand Hotel
#4I like this paradox for it's simplicity, but there's just one aspect that cracked me up.
>Suppose the hotel is next to an ocean, and an infinite number of aircraft carriers arrive, each bearing an infinite number of coaches, each with an infinite number of passengers.
hahaha.
How would we extend that?
suppose we have an infinite number of passengers, carried by an infinite number of coaches, transported by an infinite number of aircraft carriers, shoved in by an infinite number of tsunamis, which occur on an infinite number of continents, on an infinite number of Dyson spheres...
Re: Hilbert's paradox of the Grand Hotel
#5A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infinite number of complex numbers with that real component and varying imaginary components. And so on ...
In fact, there are an infinite number of infinities ...
It's turtles all the way down.
Re: Hilbert's paradox of the Grand Hotel
#6And here's a great book that explains it to children: http://www.amazon.com/The-Cat-Numberland-Ivar-Ekeland/dp/081...
Re: Hilbert's paradox of the Grand Hotel
#7A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infini…
That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
Re: Hilbert's paradox of the Grand Hotel
#8A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infini…
That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
Wait, but between n and n+1 there are also an infinite number of rational numbers, but the cardinality of all rational numbers is the same as that of integers (or that of all rational numbers between a fixed n and n+1).
So, while what the gp wrote is correct, it's not an explanation of why there are more real numbers than integers.
Re: Hilbert's paradox of the Grand Hotel
#9A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infini…
That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
I don't think the summary in the comment is correct, though. There are also an infinite number of rational numbers between any given integer n and n+1, and indeed an infinite number of rational numbers between any two rational numbers, yet the cardinality of the rationals is still the same as the cardinality of the integers.
Diagonalization is brilliant, though!
Re: Hilbert's paradox of the Grand Hotel
#10A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infini…
That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
It's a little confusing though, at least for me. There are infinite integers, and "doubly-infinite" reals. But there are also only "doubly-infinite" complex numbers.
IMO the easiest way to illustrate different, and infinite number of, infinities is to think of sets. Suppose you have N-infinite number of things, say triple-infinite to use the grandparent's lingo. Now the number of sets that you can create from these triple-infinite objects is greater than triple-infinite, and it's called quad-infinite. You can prove that indirectly. Assume that there is a one to one mapping of the original object to the sets. That means that some of the objects are mapped to sets that the objects are members of (let's call these red objects), others not (let's call them blue objects). Does the set of blue objects get paired with a blue or a red object ? Cannot be either, so we've reached a contradiction, ergo there are always more subsets of things than things.