Hilbert's Hotel

The concepts of Hilbert's Hotel involve not a paradox, as the "thought experiment" is sometimes called, but an inherent contradiction.  Many things are called possible or impossible by people who are in the grip of assumptions, and thus they wildly misunderstand both.  Some truths are necessary: they are true in themselves and could not have been any other way because anything to the contrary would still rely on these things being true.  For instance, it could not be true that there is no truth, so something (logical axioms and other necessities) is always true.  Thus, they cannot be false despite how anything in conflict with them could only be false.  The veracity of logical axioms and the fact that one cannot perceive anything at all, not even doubting that one exists, without existing as a consciousness are the examples.  Possibility is determined by whether something is consistent with these or any other necessary truths (not all necessary truths are self-evident as these are).

Anything that is not consistent with these or some other truth is genuinely impossible rather than whatever strikes someone as odd or unfamiliar.  Now, some things are logically possible in that they could have been true even if they are not.  Because I perceive grass to not be gold, it cannot possibly be true that I am perceiving it to be that very color, but, the issue of the reality beyond the perceptions aside, I could have seen red or gold grass upon entering this world rather than what I observe.  Hilbert's Hotel does not deal with something like this.  Pertaining to the nature of infinity, the idea is one of a hotel with an infinite number of rooms which are all full--however, since there is always another room in an infinitely large hotel, there would always be a place for a newcomer, or else the hotel would not really be infinite.

Infinity added to infinity is still endless at least in one direction.  If this kind of hotel was possible, one could always have the guest in one room move to the next room, such as with the person in room 106 moving to room 107.  If the number of rooms are infinite, though, there is always another room that can be taken without any current guest having to move to the next available room.  It could not be that a hotel would be entirely full and yet be capable of always accepting a new guest by having everyone move one room up/over if there is an infinite number of rooms to start with.  Since a hotel being full and yet simultaneously having additional rooms for more guests is a logical contradiction, this is impossible, though reason still necessitates that certain things follow from the premises.  It is just that the premises nullify each other because of the intrinsic axiom of non-contradiction.

A hotel of infinite rooms could be full and not full at once, rendering the idea logically impossible.  Although it is not the same as other notions related to infinity, such as the concept of the universe having an infinite number of past events, it exemplifies why some manifestations of infinity are not even hypothetically real and could not have been the case in any counterfactual universe (as necessary truths, the laws of logic would be unchanged and transcend all matter, spirit, and other existents).  An infinite sequence of past moments would never reach the present, which can be logically proven to exist.  An infinite past is thus impossible.  On the contrary, at a microscopic level, one could always reduce a log or some other object in half, given the right supernatural or technological means of doing do, because a finite distance or unit of matter could be constantly lessed in size without eliminating its existence altogether.  Hilbert's Hotel does not feature this kind of logically possible infinity.