Actual Infinity in Reality?

[SteveII]

The math is based on axioms (assumptions). It is question begging (circular reasoning) to say it is proof that they are logical possibilities. You have assumed an actual infinite in order to do further math with it. So, it gives no help to the argument that an actual infinity can exists. Therefore need we turn to something other than math:

1. You cannot get to infinity by successive addition.
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

These coupled with the fact that we don't have anything in the real world that could be an actual infinite leads a rational person to the believe that an actual infinite of real objects is not possible.

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.

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.

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?

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.