Actual Infinity in Reality?

[polymath257]

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.

1. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

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

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

4. 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:
5. infinity + infinity = infinity
6. infinity + infinity = infinity/2
7. infinity - 1 = infinity
8. infinity / 2 = infinity
9. infinity - infinity = 3

10. 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.

I want to point out that even asking these questions assumes the set theory you reject. When you talk about the infinite number of rooms of the HH, you are using the methods of set theory.

1. Not quite. The notion of 'half' of the rooms is not well defined in this context. There is a notion of density for subsets of the natural numbers, and that density is 1/2, but that is a different thing *and relies on the set theory you object to*. The notion of 'half' does not apply here unless you define what it means in this context. And *that* requires the set theory you reject.

2. An infinite number. More specifically, *in the set theory your reject* the answer is aleph_0.

3. The answer to both is 'an infinite number'. More specifically, *in the set theory you reject*, the answers are both aleph_0.

4. The answer is 3. We are using set theoretic differences here, though, *which you reject*.

5. This is correct, although you haven't defined what addition is for this context and cannot do so without the set theory you reject. But, yes, the union of two disjoint infinite sets is infinite *in the set theory you reject*.

6. As described above, the notion of infinity/2 is not well defined, although *with the set theory you reject* it can be given a meaning and the result is again aleph_0.

7. Again, your demonstration is that when you remove an element from an infinite set (which is a process you reject), you get another infinite set of the same cardinality. That is true.

8. As explained above, infinity/2 is meaningless unless you define your operations. But yes, if you take an infinite set and divide it into two sets of the same cardinalities, then both pieces will have the same cardinality as the original.

9. And once again, you are attempting to generalize from the case of finite sets to that of infinite sets. The notion of subtraction of cardinalities is not well defined. All this means in this context is that if you take an infinite collection and remove an infinite collection, the answer will depend on *how* you remove that infinite collection. That isn't a contradiction.

No, what you have shown is that the operations of division and subtraction of infinites are either not well defined (they depend on specifics of the situation) or they give results you see as paradoxical, even though they are consistent. But guess what? That happens in ordinary arithmetic. What is 3*0? What is 5*0? So is 0/0=3 or is 0/0=5? That we cannot divide by 0 doesn't mean it is contradictory to use the number 0.

So, here is a question: why do you expect the operations of subtraction and division would work the same way for infinite things as for finite things? What is the physical basis for this expectation?

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Let me ask this: do you think all those things make sense *within* the context of an infinite set theory?

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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.

 Consciousness is basic for logic? Really?

OK, what are LNC and LEM? Those are abbreviations I am not familiar with.

I would notice that you used the operations of addition, subtraction, and division. Those are not on your list, are they?