Actual Infinity in Reality?

[polymath257]

On the contrary, the fact that we do not get inconsistencies from these axioms shows there is no logical problem with them.

We know of many axiom systems that *are* self-contradictory. The theory of infinite sets is not one of them.

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.  

1. Irrelevant. That isn't the mechanism for getting infinite sets.
2. Not absurdities, just counter to intuition derived from the study of finite things.
3. Not a contradiction. Again, just counter to intuitions derived from the study of finite things.
4. Not well defined. An impossible task due to relativity.
5. Not well defined. Task impossible to do because of relativity.
6. Resolved because both space and time are infinitely divisible: see algebra and calculus.

the lack of coherent arguments against actual infinities and the fact that they are not self-contradictory is enough to show they are *possibilities*. 

Again, whether they are present in the real world is not known. But there is no *logical* issue. Paradoxes because people think in terms of finite sets, yes. But no contradictions.

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. 

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Once again, Steve, you cannot get Graham's number (which is finite) via successive additions for any operations, as fast as they could be, in the current age of the universe.

Do you think Graham's number can exist as a logical possibility?

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.

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.

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.

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.

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?