Actual Infinity in Reality?

[polymath257]

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. That's ridiculous. HH posits an infinite amount of real objects and then illustrate how it behaves absurdly when you attempt to make changes. Clear proof that an actual infinity contains absurdities that cannot be reconciled without applying a series of assumptions to make sense of it. It is in trying to make sense of it that YOU keep bringing in 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.

2. I think you literally invoked set theory in every reply. So again you use a mathematical assumption to explain why we get 5 absurd results when moving people around. That is question begging and is insufficient as an explanation let along a defense of an actual infinity. 

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.

3. No, what HH shows is that the concept of infinity is not equipped to explain real objects. The conclusions are true, yet they are absurd. Regarding "0", you cannot parody HH with 0 and show anything remotely similar. As such, the comparison you are attempting makes no progress toward "normalizing" the absurd conclusions. 

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?

4. For the very obvious reason that you need to be able to subtract and divide real objections in the real world. If you can't do that with infinite quantities, you cannot have an actual infinity of real objects. This is not rocket science. For like the ninth time, don't you find it interesting that you cannot find an article defending an actual infinity of real objects to post for us to prove your point? 

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

5. Real objects MUST be able to be divided and subtracted because they are tangible things and you HAVE TO be able to do these simple operations with tangible things. 

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 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?

Yes a consciousness is an awareness and ability to perceive reality and therefore necessary in any discussion of reality. 

6. Law of Non-contradiction and the Law of the Excluded Middle. 

7. Operations of addition, subtraction and division are results of real objects being moved about. They are not axiomatic. I don't have to assume any additional truths before conducting the operations.

 6. Fair enough. Not an abbreviation I typically use.

7. OK, you can add two infinite collections together and get another infinite collection. No problem.

If you have an infinite collection and remove a part of it, the result you get depends on which part you remove. There is no requirement that the *abstract* notions of subtraction extend to infinite quantities. The operation of addition does (even though that isn't a logical requirement either). The operation of division isn't even well defined for integers, so your claim it is required for infinite quantities is silly.

5. OK, but the fact that you can do these operations on objects doesn't mean you have to get the same result when you do the operation in a different way. And that is precisely what you were doing.

4. For the ninth time, I don't expect to because any physicist that works on this sees infinite space, for example, as a logical possibility that will need to be answered via observation, not through metaphysics. The problem with subtraction and division is that *how* you do them affects the answer. if you keep track of that, there is no contradiction. Different operations will yield different results.

3. They are only absurd because you expect infinite quatitiplay by the same rules as finite ones. That is your problem, not the results.

2. NONE of the results were absurd. They may stretch your intuitions a bit, but there really isn't a problem with any of the results that are well defined (the ones involving addition, subtraction of a finite quantity, and division by a finite quantity). Those that are NOT well defined are still consistent *if* you go to the collections themselves and don't expect a 'numerical' answer in all cases.

The *exact* same thing happens when attempting to divide by 0. You can claim that we always need to divide, but this is a good example where it is perfectly reasonable to NOT allow division. Why? Because it would potentially give more than one answer. Since 3*0=0, we expect 3=0/0. Since 5*0=0, we expect 5=0/0. Because of this, we simply do not define 0/0. It is an operation we leave undefined. The same happens, for infinity-infinity and infinity/infinity. This is *completely* analogous.

1. Every single one of those operations gave an answer, didn't they? it may not have been the answer you wanted, but an answer was given.

If you take out every odd number, you have removed infinitely many things and have left infinitely many things. If you remove all but the first 3 numbers, however (a different operation), you will have removed infinitely many things from infinite many things and have finitely many left.  Where is the absurdity? Both operations gave well defined answers. The answers were different because the specific operations were different. That's all.

I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.

At no point did you show a contradiction. You did show that certain operations are not well defined, but gave no reason to think they must be. But, at the same time, you gave the correct answers for every particular. Funny how you could do that if it is so unreasonable.