Forum:Questions Regarding Cardinal Infinities

Just yesterday I got a new book to add to my collection of mathematical books. This time it was Martin Gardner's Wheels, Life, And Other Mathematical Amusements, which I've almost read the entirety of. I found chapter 4, which was entitled "Alephs And Supertasks" particularly interesting. However, there are two questions I have.

The first question is based off of the following passage from page 35 of the book:

"Is there a set in mathematics that corresponds to $$2^c$$? Of course we know it is the number of all subsets of the real numbers, but does it apply to any familiar set in mathematics? Yes, it is the set of all real functions of x, even the set of all real one-valued functions. This is the same as the number of all possible permutations of the points on a line. Geometrically it is all the curves (including discontinuous ones) that can be drawn on a plane or even a small finite portion of a plane the size, say, of a postage stamp. As for 2 to the power of $$2^c$$, no one has yet found a set, aside from the subsets of $$2^c$$, equal to it. Only aleph-null, c, and $$2^c$$ seem to have an application outside the higher reaches of set theory."

So, if I understand correctly, does this mean the cardinality of the number of possible drawings is $$2^c$$ ?

If so, would a drawing of a single point count as a 'discountinuous curve' ? - How about a smooth line and 1,000 random points ? I'm wondering what counts as a 'real-function of x', according to the passage above.

Lastly, what is an example of a meaningful statement you can make about infinities greater than $$2^c$$ ? If you are not able to correspond such an infinity with a visual set or something easily graspable, then how can you be sure the statements you're making about that infinity are meaningful or not ? Edwin Shade (talk) 21:25, November 3, 2017 (UTC)