Does the differentiation of countable and uncountable infinities play any role in calculus?

Question

Calculus uses the concept of infinity a lot. I have never seen the type of infinity to make any difference in calculus. Are there any ideas in calculus that care what flavor of infinity is to be used?

Answer 1

Calculus talks about infinity a lot, but if you examine the definitions, you will note that they do not use actual infinities at all. Rather, everything is described finitely. Infinity is handled as the result of finite approximations. That is, we define the value of something "at infinity" to be the unique number (or other object) that makes sense because of the behavior of whatever we are defining for large finite values. If there isn't a unique value that makes sense, then we don't define a value "at infinity" at all.

You can generally equate the positive infinity of calculus with countable infinity of cardinality, as they both are approached by finite values, but the concepts are not truly the same (except when talking about the indexes of sequences or size of sets - that infinity is the cardinal infinity).

As a general rule, calculus doesn't have much to do with uncountable cardinals. Even when they do get involved, it is usually in an artificial way. For example, one occasionally sees the definition $$\sum_{x \in A} f(x) = \sup\left\{\sum_{k=0}^n f(x_k)\ :\ \{x_k\}_{k=0}^n \subseteq A\right\}$$ defined for $f\;:\;A \to \Bbb R_{\ge 0}$ where $A$ is a set of any cardinality. But you can show this only converges when $f = 0$ except on a countable subset of $A$.