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Re: Infinity

Having watched the video, I wouldn't say that all the four paradox examples are based on the idea of infinity = infinity + 1. That's only true of the first example, Hilbert's Hotel, and that only applies specifically in the case of one new guest turning up. The second example is based on a container like object having an infinite surface area but a finite internal volume, the third example is based on a mathematical point having zero area associated with it, and the fourth example is based on the idea of playing a game organised by a casino which has an infinite supply of money.

Some more details of the Hilbert's Hotel example are given in this Wikipedia article:

You can go further than just having one new guest turn up. If n new guests turn up, the existing residents could be moved from room 1 to n+1, room 2 to n+2, etc. You can also handle an infinite number of new guests arriving by moving existing residents to the closest even numbered rooms and then putting the new guests in the odd numbered rooms. The Wikipedia article also outlines a numbering scheme for how to accommodate an infinite number of coaches each containing an infinite number of new guests arriving at the hotel.

I think the most interesting of these infinity-related paradoxes is one I was confidently expecting to be in the video, but isn't included, a paradox called the 'coastline paradox', which is described in this Wikipedia article: