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 a few 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 ? 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 , no one has yet found a set, aside from the subsets of , equal to it. Only aleph-null, c, and 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 ?
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 ? 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)
- I have not encountered anyone allowing curves to be discontinuous, and because of that I am unsure of what exactly Gardner considers to be a "curve". Usually, a "curve" is defined either as a continuous function \(f:[0,1]\rightarrow\mathbb R^2\) or the image of such function (think of it this way: the latter is just some doodle on the plane which we can draw with a pen, whereas the former also encodes how we drew the doodle, for example how fast we were moving the pen). If we throw away the continuity assumption, then for the latter definition, a "curve" can be literally any nonempty subset of the plane - for example, the point is the result of keeping the pen in one spot (so it is in fact a continuous curve), I'm going to leave it to your imagination to think of how we might draw your line and points (this one will be discontinuous). Using the former definition of a curve just adds another complication to that.
- As for "real function of x", I believe this simply means all functions from \(\mathbb R\) to itself (x denotes the variable, but it could be any other letter or symbol).
- I am not sure what you mean here with "meaningful statements". Even if we agree that there are no sets of those higher cardinalities which are easy to visualize, such sets still exist, for example if we consider the set of all sets of functions \(\mathbb R\to\mathbb R\). An example of a statement about such larger cardinalities is the Cantor's theorem, that a power set of such a set has an even larger cardinality (sure, it holds for smaller sets too, but it doesn't make it invalid for large sets). We don't really care whether the statement is "meaningful" or not, whatever you mean with it. We care whether it is true or not. LittlePeng9 (talk) 22:24, November 3, 2017 (UTC)
- Thank you for your answers. I was wondering by "meaningful" if there is anything useful you can say about a large cardinal infinity other than that it's bigger than the last. Edwin Shade (talk) 00:50, November 4, 2017 (UTC)