Actual Infinity in Reality?

[polymath257]

1. One standard way to show the impossibility of something is a proof by contradiction. If you assume the existence and derive a contradiction, you have established the non-existence. But, in spite of many attempts to show a contradiction in the notion of actual infinities, no such contradiction has ever been found.

1. It is utterly confusing to me why you can't see that your entire #1 is exactly what I have done with my list of 6. 

2. What conflicting answers? Be specific. There are two notions of size relevant to sets: containment and one-to-one correspondence. They are different ways to describe size and yes, they can give different answers. That isn't a contradiction any more than the fact that volume and mass can give different answers to the question of 'how much?'. All that is required to resolve this 'absurdity' is more precise language.

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. 

The impossibilities of Thomson and Ross-Littlewood are not in the notion of infinity, but the fact that the activities required cannot be done because of relativistic effects.

3. That's nonsense. Relatively has nothing to do with Thomson's two minutes of light switching or Ross-Littlewood's 30 seconds of ball tossing. 

Zeno's paradoxes were *solved* by the introduction of infinities! The infinite divisibility of both space and time nicely solve ALL of the Zeno paradoxes.

4. Except that the Dichotomy paradox example of moving a distance can never start because you always need to traverse the first fraction of the distance--but that fraction is infinitely small. Yet we reach our goals with quite regularity in the real world. So, it would seem you can and cannot traverse an infinite number of points. A contradiction solved by deciding that infinities do not work the same in division as in multiplication, addition and subtraction. It illustrates another aspect of infinity does not translate well into the real world. 

3. Well, one of your objections to the notion of an actual infinity is that it cannot be counted to (which is, truthfully, irrelevant). Neither can Graham's number. So why do you accept one as a possibility and not the other?

What? Do you really think that Graham's number has the same properties of an actual infinity? Of course it is metaphysically possible to count by 1 to Graham's number.

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.