Bound Infinity

I was talking with a friend recently, and the topic of infinity came up. I referenced a Veritasium video, regarding the hotel with infinite rooms running out of rooms. The specific scenario had to do with a bus showing up with an infinite number of individuals that had letter designations, filling all rooms. Then an additional person with a diagonal cross section of the letters was added, who would be unique from the infinite rest, but there would be no room in the infinite rooms for the additional person.

My friend stood against me in my explanation, and interposed the logic presented in the video, saying that there was trickery involved, like a shell game, because infinity cannot be exhausted. Today I realized that he was right, not specifically in this scenario but in general, and I was able to see the difference between the scenario presented and "true," or conceptual infinity. Much like I have described here previously that zero is not nothing, but rather the mathematical representation of nothing, infinity in mathematics differs from truely infinite in the same way; it is a concept bound by numbers. In both cases, these bindings are so sleek that it is easy to see why most do not consider them to not be there at all, but the scenario presented in the video clearly demonstrates how binding infinity to numbers does materially change infinity, and limit infinity (binding infinity). 

Essentially what the Veritasium founder, Derek Muller, did, was take conceptual infinity, bound in a single way, by defining it as a string of numbers, and then introduce it to an infinity that was bound in more than one way, with letter designations, also still leaving the requirement of satisfying an overall infinite numerical requirement, but adding an additional identifier in each place, with the letters. I realized that this struck my friend as incorrect, and I could not figure out initially how best to explain how it wasn't, because most people (including myself until now) do not register numerically infinite as a bound form of conceptually infinite.

As I considered this further, I realized that increasingly bound infinities could continue to overload less bound infinities in this way. Each additional identifier, within what a single integer would normal occupy, would create additional complexity (A instead of 1, A1 instead of 1 or A, the shading of a QR code, etc.). I am having trouble thinking this all the way through, but it could be that an infinite number of these additional bindings (rules), could exist, in a logically advancing way, so that one would ultimately have an additional layer of infinite scenarios to assess. Ultimately though, despite the exhaustion of each bound infinity within the string, conceptual infinity would remain unscathed, as the protecting the one conceiving of this persists, and will continue to do so forever (or, at some point cease, at which point the whole examination becomes meaningless to them, from their perspective). This has a beautiful symmetry with nothing, conceptually, and continues to spark my imagination even now. It does feel like the closest that one gets to perceiving conceptual infinity is from their own perspective, which is to say that conceiving of numerical infinity from a singular perspective is, in and of itself, glimpsing conceptual infinity.