Actual Infinity in Reality?

[SteveII]

1. Again, question begging. By axiom, you assume something exists. That cannot be then used as proof of that thing existing. You did not get to the assumption by logic, therefore you cannot say that it is logical.  


2. What?!? Conflicting answers (Hilbert, Galileo), impossibilities (Ross-Littlewood, Thomson), and obviously false (Zeno) is not just "counter to intuition". Your bar is set really, really low for metaphysical impossibilities. Your reasoning is that we don't assume mathematical non-logical axioms--therefore we can't make sense of the paradoxes. That is clearly question-begging. 


3. I have no idea why you might think that Graham's number has a logical problem. It has none at all. Ironically, there are an infinite amount of numbers that could not be counted to in any age of any universe. The fact that you think this is a point is puzzling.

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.

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.

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.

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.

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.

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Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

Start with these. 

1. You cannot get to infinity by successive addition. That means the actual infinite cannot be built--it must already exist. 
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

and then give me one good reason why I should ignore all these paradoxes and absurdities and believe that an actual infinite is possible. 


BTW, you are missing the whole point of Hilbert's Hotel:

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.