The Infinite Hotel Paradox - Jeff Dekofsky
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when explaining countable infinity and why infinity behaves differently from ordinary numbers, and as a friendly introduction to Hilbert's paradox.
Recommended by 3 people across 4 different months, 2014 to 2018.
What it is
A six-minute TED-Ed animation, narrated as a lesson by Jeff Dekofsky, that walks through Hilbert's Hotel thought experiment. It shows how a fully booked hotel with infinitely many rooms can still take one new guest, 40 guests, or an infinitely full bus of guests.
What people said
Recommends goblin GONE · 2015-03 · Mathematics Stack Exchange · on “If $dx$ is an infinitesimal, can't you list all real numbers as the sequence each whole number times dx?”
@veta, check out this [LINK] video to learn a few basic facts. But note that some of the stuff that is said after $8:30$ is kind of incorrect. A little bit. This [this video] is also good stuff.
Recommends The Integrator · 2018-03 · Mathematics Stack Exchange · on “The "Paradox" of Hilbert's Hotel”
The number of rooms in Hilberts hotel is a countable infinity and is completely filled with an infinite number of guests . Just think of it as there being a $1$ to $1$ correspondence with the natural numbers {$\,1,2,3,4,5,....$}and think of the guests being numbers, So the hotel is always completely full since each number staying there is always matched one to one to a room. The reason you cant put a new guest at the last room is because there is no last room, there is always an room after the so called "last room" . But you can make space for a new guest if you move each guest to the next room ie $1 \rightarrow 2,2\rightarrow3,3\rightarrow,......$ie $n\rightarrow n+1$. I'll leave a link to
Recommends Rhys Hughes · 2018-05 · Mathematics Stack Exchange · on “Can one infinity be greater than other?”
No. Infinity is a concept, not a number. Therefore, the limits, while one accelerates at a faster rate, both have the same endpoint of infinity, and therefore are equal. Have a look at this [this video] for a clearer explanation of why $\infty$ is not a number.
Recommends Rhys Hughes · 2018-06 · Mathematics Stack Exchange · on “Does infinity belongs to the set of all real numbers?”
I'm reminded of this video [this video] when I see this sort of question. Essentially, $\infty$ is just a concept . It doesn't behave in the same way that the common real number does. To see this, consider the two equations: $$x=x+1$$ $$x=2x$$ Notice that apart from the second being satisfied by $0$, there exists no number which satisfies either of these. However, due to the arbitrary nature of the concept $\infty$, it solves both, as explained in the video.
Cites Johannes P. · 2014-02 · Mathematics Stack Exchange · on “Is it faster to count to the infinite going one by one or two by two?”
+1, thorough explanation. As for mapping to the natural numbers you might also encounter the term bijection [LINK] . And as for paradoxes, you might enjoy Hilbert's Hotel [this video] .