My background: I have a bachelor's CS degree and have never taken anything beyond part of a first course in abstract algebra - no real analysis or complex analysis. I learned about higher cardinalities than $R$ in my automata course.
The inverse of addition works on integers.
The inverse of multiplication forces us to get to the rationals, but that's still the same cardinality.
The inverse of exponentiation forces us out of the rationals and into complex numbers, but we're still at integer cardinality because it's algebraic.
Consider $a^x$ is transcendental at least for $a>1$ and when given imaginary variables (at $x=0$ it will look like a sine wave, probably different period than $sin(x)$, which is associated with e^x, along the imaginary part of y axis) needs a real cardinality.
Therefore, $x^x$, the next level up from exponentiation is going to be at least real-valued in cardinality, as well. By simply increasing the "level" of my operation, I bumped my way out of the integers.
My intuition says if I keep bumping up my operation, eventually I may have a function such that if I need a cardinality greater even than the reals to smooth it out.
Is there a "jagged" real-valued function that is "smooth" in cardinalities greater than the reals?
I'd say something is "smooth" if it is not "flat" and if it always approaches being "flatter" as you "zoom in". Flat would mean minimum distance and flatter would mean approaches without overshooting. Like $sin(x) * e^x$ and $e^x$ both "approach" 0 (in the the range necessary to contain all values towards infinity strictly decreases) as x grows arbitrarily negative, even though in the first case, it "overshoots" and crosses the $x$-axis.
Just like the reals are in between the integers that fill in the gaps of some functions, I wondered if there were numbers in between the reals to fill in the gaps of some functions, but I decided to try to word it a little more precisely.
In response to "How can a function on the reals be extended to a completely different set? And how do you measure it?":
Consider all finite non-empty sequences of $1$'s and $0$'s that start with a $1$, call it $S_1$. This is all positive integers base-2.
Now consider all proper subsets of that set, call it $S_2$.
Now consider $f_1$ that takes in an element of $S_2$ and returns a string whose $i$th digit is $1$ if the base-2 representation of $i$ is in $S_2$, $0$ otherwise.
Now consider the elements of $S_3 = \{ (s_1, f_1(s_2)) | s_2 \in S_2, s_1 \in S_1 \cap \{0\} \}$. This will correspond to all non-negative real numbers. The first thing in the tuple will be the part of the number to the left of the decimal. The second thing in the tuple will be the decimal part.
Is $S_1$ a subset of $S_3$? We can map $S_1$ to $S_3$ by saying elements $s_1$ in $S_1$ correspond to the elements in $S_3$ that have $s_2$ as the empty set and have $s_1$ as the first thing in their tuple. In other words integers get mapped to the reals that have all $0$'s as their decimal portion.
Now consider all proper subsets of $S_3$, call it $S_4$.
Now consider the elements of $S_5 = \{ (s_3, s_4) | s_3 \in S_3, s_4 \in S_4 \}$. This is our greater-than-the-reals set (non-negative reals), and we say the reals are a subset of this set by saying they correspond to the elements of $S_5$ where $s_4$ is the empty set.
I can give a partial definition of measuring by assuming $s_3$ is greater than $t_3$ and that $s_4$ is a superset of $t_4$ and say their distance is $(s_3 - t_3, s_4 - t_4)$. I don't know what to do if $s_3 > s_4$ and $t_3 \not \supset t_4$ because you need some way to "borrow" like in basic arithmetic if you try to remove a set that isn't there.
