Actual Infinity in Reality?

[SteveII]

2. Your answer to Hillbert's hotel is that "with infinite sets...". I have shown conclusively that any argument that contains the words "infinite set..." is question begging. You have assumed what you are trying to prove. You need to look that up if you are fuzzy on that. 

1. And I am pointing out that you failed to demonstrate a contradiction.

2. YOUR claim was that the Hilbert Hotel leads to absurdities. WHAT are the supposed absurdities? Since we are talking about the HH, of course, we are assuming an actual infinity. But that is how proof via contradiction works: you assume the result you want o show is false, then you arrive at a contradiction. You have not done so.

3. Two switch at more than a certain rate would require the switch to move faster than light. That is impossible. Similarly, in the Ross-Littlewood scenario, we cannot move the balls faster than light, so cannot remove and  replace faster than a certain rate.

Hmmm...it seems to me that you have never given your metaphysical axioms....care to show which assumptions you are working with? Are you assuming that everything must be finite (and hence begging the question)?

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Again, what *specifically* is absurd in the HH? Give details.

Okay, without appealing to infinite set theory (because that would be question begging) tell me why this all makes sense:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

How many people would be in the hotel if the guest in #1 checked out?

If everyone in odd number rooms checks out, how many checked out? How many are left?

Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

Conclusion: the idea of an actual infinite is logically absurd.

Regarding metaphysical axioms for this discussion, just the basics. Existence, consciousness, the Law of Identity, LNC, and LEM. Notice these axioms are self-evident and can not fail to be true and therefore are not in the same class as the math-specific Axiom of Infinity.