Not at all. I may produce scenarios in which a theory that makes use of infinite quantities is the most explanatory, but I don't claim that they're actual. I'm asking a metaphysical rather than a factual question.
But to have it said: If we are no longer disputing the impossibility of infinite quantities, and it is recognized that no satisfying argument has been provided to that effect, then I am happily unconcerned with other issues. However, I'll still respond to the questions below.
You mean, suppose that it is impossible for there to be infinite quantities, and then you ask if there is any other way in which evidence can be proffered for the existence of infinite quantities? I doubt it, but if it can be shown that infinite quantities are impossible, then I will be satisfied in the first place.
Yes. Why do I accept the existence of atoms? Because the atomic theory is that which best explains and predicts behavior which I can observe. I have never observed an atom, but the atomic theory has too many theoretical virtues to justify rejecting it. It is (relatively) simple, lends to calculation, predicts events accurately, accurately suggests ways to manipulate the world around us, and so on. In fact, simplicity was the original reason for the popular adoption of heliocentrism. Geocentrism was never disproved, probably until we launched a person into space. While Ptolemaic geocentrism could accurately predict the location of celestial bodies, for an observer at a fixed parallel, when shifting parallels the calculations were no longer accurate. Trying to come up with Ptolemaic models of celestial positions and orbits became computationally nightmarish, when Copernicus's incredibly simple model lent to easy computation with just as much accuracy, and so it was adopted. I don't believe either of these theoretical changes are philosophically suspect, in spite of not having been supported by direct observation.
Likewise, in general, if a theory has such theoretical virtues, then I come to believe the theory without insisting on direct observation. Thus if a theory has these virtues and makes essential claim to an actual infinity of objects, then I will believe in the theory and thereby believe in an infinity of objects.
So long as the theories are not self-contradictory, yes, I take them to be possible in that sense. I take the claim that there is a person on the roof of my building to be equally arbitrary in that sense, and equally possible in that sense. Some of the arbitrary claims will sound fantastic because we have never seen or heard anything like it before, some of them will sound mundane and more plausible because we have seen similar things, but that doesn't affect the fact that they're equally unsupported by observational evidence and so--in the sense you're using the term "arbitrary" here--equally arbitrary. And fittingly, I don't believe in the existence of Zeus, and I don't believe in the existence of an infinite quantity.
Now, if somebody were to claim to have evidence for the existence of Zeus, I would be more suspicious since there seems to be so much evidence against it: What's he been doing for the last couple thousand years at the least? Where's he been? Moreover, the story of Zeus seems to be too easily explained by child-like fantasies and desperate attempts by ignorant people to explain the world. That is to say, I would be more reluctant to accept the Zeus hypothesis because I already have another one that is working quite well. But if, some how--say by direct observation, or by theories which make essential use of the Zeus hypothesis to predict all events better than current theories, etc.--there were better evidence for Zeus than against him, then I would believe that Zeus exists. Infinity, on the other hand, has none of these unpleasant properties. Current theories, as far as I can tell, are completely agnostic on this account and so there is no evidence against the existence of infinite quantities. Moreover, it doesn't seem like some infantile attempt at explaining anything. So far, it's no explanation at all in the first place--there is no attempt at explaining any fact by reference to infinite quantities. The hypothesis of an actual infinity does not claim to actually solve anything, just yet. But if it did, and if there were good evidence and theoretical virtues, then it would have a case worth listening to. And as we seem to have established, there is no reason why this theory would violate the Principle of Identity, and so there would no reason to reject it out of hand.
Why is that problematic? There wouldn't be an end-point. So what?
Reference? And why can you not talk about "greater than"? Sure, the natural numbers are infinite, but you can still pick some two natural numbers and compare whether they're greater, less, or equal to each other. Comparing some natural number to the number infinity itself? Well, here you'd have to somewhat expand your notion of "greater" and "equinumerous", or you'd be right, there would be no sense in comparing them. But if your notion of "greater" is redefined by the notion of mappings, as we started with, then infinite sets will be greater in quantity than any finite set.
Well then, what's the axiom? You can't just say it's an axiom without naming what axiom you're appealing to. And if you say "identity", then you have to say how identity precludes infinity. Because as far as I can tell, there is no connection whatsoever between the axiom of infinity and identity. If you bring up the point about boundaries, then you again have to say why it is that lacking a boundary on quantity is impossible.
I.e., you have not said anything new here, and only suggest a rehearsal of a long conversation that has already played out, and ended with no good argument against the possibility of an infinite quantity. This is intellectually irresponsible, since it attempts to suggest that the case has been made to satisfaction, for the impossibility of an infinite quantity, when no such thing is even remotely true.
This point has been made before, and refuted before, unless you can come up with a counter-counter-argument.
By saying that the natural numbers are infinite, I am merely taking this to be a piece of our mathematical rules. It's not circularity, but statement of definition. To have the size of the set of natural numbers is just to stand in an bijection relation to the natural numbers, which has the properties you mention above. But sets do not grow by thinking up new members--sets are determinate collections of objects. So in sets, there is no such thing as the possibly infinite. There is only what is actually in the set and what is not.
However, as stated a few posts into the discussion, even if we amend our mathematics to accommodate a notion of possible infinity, I may simply re-phrase my question: Is it possible or impossible for there to be, for every set of natural numbers, an injective map from that set to some set of disjoint physical objects? If so, then I would take this to be my definition for there being an infinite quantity.
I've responded to all this before.
He argues that the universe was caused, from the assumption that it has a beginning time. I don't know if he argues, or how he argues, the acceptance of the assumption, but it's not the same as arguing against an actual infinity. If you know of some such argument that he does employ, please reproduce it here because it is not an article of common knowledge, or even of widely known philosophical knowledge.