17:30 to 18:52 – 1 person arrives when the hotel is full.
Solution: all the people move 1 room position.
18:52 to 20:00 – 66 people arrive when the hotel is full.
Solution: all the people move 66 room positions.
20:00 to 21:20 – a bus arrives with countably infinitely many new arrivals.
Solution: each person moves to the number twice their current room number.
21:20 to 24:40 – an infinite number of buses arrive, each carrying an infinite number of people.
Solution: as a paper exercise, re-arrange the hotel rooms into pyramid shape (triangular),
24:40 to 27:55 – explaining how all the positive rational numbers (i.e., can be represented as a fraction) are countable – includes illustrations.
27:55 to 33:05 – the diagonal argument – demonstrating how the set of real numbers in the interval 0 to 1 is not “countable” – “proof by contradiction”.
33:05 to 34:02 – the rationals can be thought of as precisely the collection of decimals which terminate or repeat.
34:02 to 34:25 – appreciating how the rational numbers compare to the real numbers, and therefore why the “reals” are not countable, but the “rationals” are.
For a few additional details follow this link: Hilbert’s hotel – The sins of the 3 dots “...”
Such a hotel would take an infinite amount of time to build, and so could not be completed in reality – never mind the fact that planet Earth would be infinitely too small, i.e., infinitesimal from the perspective of the hotel.
Or if we simply assume it does actually exist – the continuity paradox of distance means the last room is disconnected from the 1st...
The hotelier, in reception at the time, was unable to walk to the far end of the hotel to issue an eviction notice to several guests (members of the jammy group) that hadn’t paid their bill, because he couldn’t make headway towards them... he walked, jogged, and even sprinted, but all to no avail.
When some of the members in group jammy tried to pay their bill via internet banking, the radio waves transmitted from their laptops didn’t make headway towards the wireless router in reception, despite travelling at approximately 186,000 miles per second.
Numerous members of the jammy group complained that the electricity wasn’t working in their rooms, so the hotelier reset the master circuit breakers by operating the reset switch connected to the control panel, mounted on the wall just outside reception, adjacent to the substation that powers the whole building...
The transfer rate of movement between electrons (electromagnetic wave) travels at approximately 50 to 99% the speed of light (186,000 miles per second; MPS), depending on the cable’s velocity factor (VF) – therefore any 240Hz alternating polarity transfers current at approximately 186,000 ÷ 240 = 0 to 775 miles per alternation (assuming 99%), infinitely short of reaching the jammy members consumer units.
Even a frequency of only 1Hz wouldn’t produce the infinite velocity required – the alternating current (AC) would simply travel a distance of 186,000 miles per alternation. Although 0Hz (direct current; DC) would indeed produce an indefinite accumulative distance, the electricity still wouldn’t make headway towards group jammy.
Consider yet again, the lack of continuity between the hotel's transformer and the jammy group – no measurable potential difference exists to produce an electric current in the first place.
The hotelier suspected something wasn’t right, so he picked up the phone and called the guest in room number infinity + 0, and the guest in room infinity + 1, but the phone signals were unable to make headway down the telephone lines towards the jammy members rooms.
The hotelier, feeling extremely concerned by this stage, fetched out the blueprints for the hotel – he noticed that room infinity + 0 neither had relative positional coordinates to reception, nor indeed to room infinity + 1. In fact, he realised that all rooms beyond room infinity + 0 must be in the exact same location as one another, and therefore give rise to infinite density – this petrified the hotelier, who understood that infinite density put planet Earth in danger of being consumed.