“The infinite sequence (.3, .33, .333, .3333…) converges to the limit 1/3″, which is another way of saying “We can make an element of (.3, .33, .333…) as close to 1/3 as we wish".
Not exactly. It means that most of the elements of {.3, .33, .333, …} are close to 1/3. Here “most” means “all but finitely many”, and “close” means “within any predetermined positive distance”. That’s how you get that limits are uniquely determined. The terms of a sequence can’t all be clustered around a and all be clustered around b.
“As a side comment: if 3/10 is not 1/3, and 33/100 is not 1/3, at what point does another digit make it exactly 1/3?”
There isn’t one. The limit of the sequence is not (generally) a term of the sequence. See comment #9 (my response to “some bla guy”).
“This is a bit like Zeno’s paradoxes, which have not been fully resolved.”
Sure they have, in large part by the limit concept. (If they hadn’t been resolved, even Newtonian physics wouldn’t be possible.)
“The meta-point is that we can make that sequence as close to 1/3 as we need, which in the real number system means they are equal in the limit.”
Limits are unique in the hyperreals as well. You might have infinitesimal separations, but you also have infinitesimal resolving power (if that makes sense). Be careful here: a lot of “standard” sequences with limits, such as {.3, .33, .333, …}, don’t have limits in the hyperreals unless you make some canonical extension to a hypersequence; if you do that, the extension will still have the original limit (in this case 1/3).
“The real number system may be “less real” because it’s more limited than others.”
Not so much. See, the construction that Robinson applied to the reals to get the hyperreals can also be applied to the hyperreals. If we call the result the hyperhyperreals, well, we can apply the construction again, to get the (hyper)^3-reals, and so forth. Each is “less limited” than the previous, but none of these can be the “real” system, because each is “more limited” than the next. But really, none of these is more limited than the others, because the same expressible facts are true in all of them.
To me, the “real” system is the simplest system which easily models the phenomena we’re interested in—which in this case is the ordinary real line.