Infinity is a bad abstraction. Mathematicians make them too.
Also I disagree that thinking about infinity is useful when thinking about large numbers. That seems to me to be a fallacy. Any finite large number is inconsequential, and might as well be a 'small' number for any practical purposes, compared to any infinity.
As trivial examples, if we reject this definition of cardinality, we can no longer speak of the natural numbers as a set (nor therefore even finite subsets of them), we would lose key set-theoretic definitions of both natural and real numbers, we could not talk set-theoretically about the domain/range of most interesting functions, etc. Honestly, that would be a much more confusing mathematical world than one with Hilbert's hotel or Banach-Tarski spheres.
Also, check out Fourier transform and try to rewrite it without infinities - https://bookdown.org/vshahrez/lecture-notes/fourier-transfor... . Without fourier transform we wouldn’t have mp3, and the whole field of signal processing if I’m not mistaken.
Without signal processing we would not have long distance digital communication.
But you can also make calculus without limits, using explicit definitions of infinitesimals (called non-standard analysis or non-standard calculus), and in some ways this is simpler to use.
You can approach it either way. As limits, or by extending the number system with particular consistent definitions of infinitesimals and infinities.
In practice using infinitesimals is often what people actually do when using calculus anyway, even when mentioning the limit in a sloppy hand-wave in the background, so it's good that non-standard analysis shows that the simpler method people actually use is usually logically coherent and gives the same answers as limits.
And even if some math concepts can't be applied to anything practical yet, we never know what will be used to formulate another theory about the stock market behaviour or processes inside the black hole.
Coming back to your question, here's an example: many math constants (pi, e) and functions (sin, cos) can be defined as infinite sums (series). They converge, getting closer and closer to the limit, but they never achieve it.
Using these series it's possible to calculate (in a calculator, Excel, Quake or CAD) values of these functions with any required precision.
Infinite series, where an infinite sequence is summed (or multiplied, or ...). Many useful things fell out of reasoning about entire infinite sequences as if they are a single unit. I like this video about a method Newton found how to calculate pi efficiently, from playing with infinite series: https://www.youtube.com/watch?v=gMlf1ELvRzc
Again it's possible to define everything about infinite series in terms of limits, but sometimes it's just a more useful thinking aid to think of infinite series as themselves. In a way, it's like dropping some unnecessary syntax sugar (the limits) as you realise you can still do useful things without it. And in fact it's possible to reason consistently about some kinds of divergent infinite series, where limits don't work but they are still logically consistent.
Complex exponentials. The discovery and proof of Euler's formula, eⁱˣ=cos(x)+i⋅sin(x) comes out of reasoning about infinite series, showing that it's a consistent and useful definition of a complex exponential. Nowadays we use these exponentials to calculate all sorts of things in engineering and physics, we just take them for granted because they work and everything fits together consistently.
Projective geometry, for example used in computer graphics. There is the concept of "point at infinity", which is actually represented by real values in a vector but in some respects you can think of it as representing "x/0" in the context of that geometry, and finding that further calculations using it still work out consistently to non-infinite results. Including the point at infinity as a value simplifies the system. Unlike with natural numbers where "infinity" is a complication that obeys different rules to numbers, the point at infinity in projective geometry removes edge cases and simplifies the rules.
Elliptic curve cryptography also uses the "point at infinity" concept which arises from a geometrical or algebraic interpretation of the elliptic curve operations, and like in projective geometry it's a useful and consistent value to include, which is done because it simplifies the system to include it.
It seems we're using the 'hey-everybody-move' as a buffer to accommodate new guests.
Two questions:
Is it ok to think because there's an infinite amount of moving, this buffer is infinitely large?
Why can't we tell guests to just keep moving to random rooms? Would that also solve the problem? If you say no - random moves don't work - then I would propose that some of those random moves would fall into the outcomes of Hilbert's formula. Is that not sufficient?
It's an unphysical model, that only exists in abstract mathematics.
Any idea on if directing guests to move randomly would work as well as directing them to move in patterns?
For example you can assign an integer to each guest, and an integer to each room, and the "move" is just the function: guests --> rooms.
As long as this function is injective, then you do not put more than 1 guest in each room.
Then this is just the observation that you can put every guest in a room and still have some empty rooms (rooms not in the range of the function). For example, the deep and paradoxical function f(n) = n+1 corresponds to an assignment of every guest that used to be in room n to room n+1.
I guess if you want, you can think of a description of the function as a sequence of "moves" with people in an actual hotel, but then since the domain of the function is infinite you have to think of an infinite "number" of moves. Then you add time to the mix and carpet hallways or whatever, you start getting lost and saying the whole thing doesn't make sense, but that's only because you are stretching these metaphors beyond the breaking point when what is really being described by the "paradox" is that you can have an injective function from an infinite set to itself that is not surjective.
By the way, once you realize that, for example, when dealing with positive real numbers there exists functions of the form: f(x) = x + 1, then you see that this "hotel" doesn't even need countably infinite moves, it can have uncountably infinite moves. Just please stop trying to worry about the weight of the hotel or how the HVAC works and getting all tangled up in stretching metaphors past their breaking point.