Everyone moves at the same time.how can you just shift one passenger to another room when they are all occupied infinitely?

Is there any guest that can't move? No.

So what's the problem?

You seem to imply that 'infinite' means 'everything'. That's not what it means.On what logical grounds can you say infinity some how implies an open ended situation when you have already claimed all infinite rooms are full?

one possible slant:

The manager says to the incoming guests, "You can't be looking for a room, as you already have one...after all are you not a part of the infinite guests to begin with? So what room do you already have?the rational being explored is:

Infinity may be "never ending" but it is always a complete series [ with in a given scenario ].

In this case we have an HOTEL with infinite rooms and infinite guests. [ a "complete" yet "never ending" scenario ]

or;

Since when has an infinite number of guests not included all possible guests?

For example, there are infinite even integers, right? But that does not mean that

*all*integers are even.

So, we can have infinite guests in the hotel, and still guests outside.

No, there are infinite guests in the hotel.this also suggests a contradiction or more a hypocrisy.

We can have an infinite number of rooms but a less than infinite number of guest....

"We just move them along one..." implies to me that whilst the rooms are infinite the guests aren't.

If the guests aren't infinite, then they must be finite. So how many are there?

Correct. But the whole point of Hilbert's Hotel is that we can manipulate that association.(R1,G1) + (R2,G2) + (R3,G3) ....

whereby the rooms and guest are directly associated infinitely.

All possible rooms and guests are accounted for. [in this single HOTEL]

Look at this direct association:

(R2,G1) + (R3,G2) + (R4,G3) ....

All previous guests are accounted for, but R1 is now a spare room.