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?
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.
Perhaps you can give a specific example of some function you have in mind.
You use the phrase, "this difference, the limit function minus f(a) itself ..." and asked if this is a cardinality. But it's a NUMBER, not a set. [Numbers are ultimately sets, but not in this context!]
For example let $f(x) = 0$ if $x \neq 0$, and $f(0) = 47$. In other words when you input anything other than $0$ into the function, you get back $0$. But when you put in $0$, you get $47$.
Now in this case the limit of $f$ as $x \to 0$ is $0$. But $f(0) = 47$. So the difference of the limit of the function at that point, and the value of the function at that point, is $47$. It's a number.
Does that make sense? Remember, a function doesn't even need to have a value at a given point in order to have a limit there.
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3) The set of real numbers, being ["absolute" is redundant and informal, I guess] a continuum, are represented by what cardinality?
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].
These are very deep waters. Nobody has any idea what is the cardinality of the continuum. We know it's $2^{\aleph_0}$, but
nobody has any idea what cardinal that is.
You asked a great question, that's a fact.
4) Can the set of real numbers have a bijection onto any finite surface?
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.
Is anything I wrote helpful? I'm not sure what is the intent of your questions.