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I can't say I have spent years studying infinite.
The Paradoxes presented in this video all seem to have at their base the idea of infinite + 1. It is a tempting notion I suppose. Exploring the edges of math is never a completely terrible idea. I just see a leap of irrationality in ignoring the conundrum that infinity = infinity + 1. As soon as you start talking about infinity +1 you are ignoring the definition of infinity and are wandering around in a la la land.
Complex numbers are a wonderful too for dealing with electronics. I always saw it as a way of dealing with orthogonal concepts.
Does infinity + 1 give us any interesting tools?
When someone starts talking about this, should I listen intently or run before he manages to pick my pocket.
I have seen one idea that resembles Hilberts Hotel. In library science, instead of reshelving books in a particular order. Just scan the book and the location you put it. When someone requests the book, the robot grabs it from the location. The only problem happens when the database blows up. Connecting that to Hilberts Hotel is a stretch.
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:
It was first noticed by a scientist called Richardson in the early 20th Century, who observed that published data for the length of a border between various countries often showed significant differences from one source of data to another. "The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines." Basically the smaller the length of the 'ruler' you use to measure the length of a coastline, the greater the length of the coastline appears to be, because of an increasing number of random nook and cranny type details that get included in the measurement. The measured length would tend to infinity as the ruler length tends to zero. But from the point of view of somebody travelling along the coastline, it obviously has a finite length.