Hi Loren. I'm not sure exactly what you're getting at. Can you provide some context? Limits in the real numbers are defined by quantifying over uncountable sets. When we say, "for every epsilon ..." and then give the rest of the modern formal definition, we are implicitly quantifying over uncountably many potential epsilons. This has philosophical and historical implications when we are talking about how the limit concept replaced the old idea of infinitesimals. But I don't know if that's what your question is getting at, or if you have something else in mind.2) Does the limit function approach f(a) infinitesimally, e.g. uncountably in decimal form? Might this difference, the limit function minus f(a) itself, be represented by a set of infinite cardinality, since the limit function is actually arbitrary?
Well, the cardinality of the continuum is the problem that's been driving set theory since Cantor first asked the question. Our usual axioms don't tell us the cardinality of the real number continuum, and the hunt for new axioms hasn't resolved the question. Some think that asking about the cardinality of the continuum is not a meaningful question. Other set theorists have esoteric approaches such as Ultimate-L and the set-theoretic multiverse. [Those are ideas I've read a little about. I would not want to give the impression I know what they mean].1)
3) The set of real numbers, being ["absolute" is redundant and informal, I guess] a continuum, are represented by what cardinality?
Does this relate to your previous question about space-filling curves? They are generally surjections but not bijections. They hit some points more than once. There's a bijection between the real numbers and the unit square, if that's what you mean by a finite surface. You can get a bijection of the unit interval to the unit square by interleaving the digits of the decimal expansions of the coordinates of the points in the square. Then you can biject the unit interval to the reals. And you have to wave your hands at the .4999... = .5 problem. There are only countably many of those pesky things so you can either ignore them or sit down and figure out how to deal with them explicitly.4) Can the set of real numbers have a bijection onto any finite surface?
But limits in other contexts quantify over other sets. For example: the real numbers are limits of sequences of rational numbers. In this context it makes no sense to have any irrational epsilon because they haven't been defined.Limits in the real numbers are defined by quantifying over uncountable sets.