
Re: Cardinality of turning wheel
Posted:
Mar 7, 2013 12:03 PM


Don Kuenz wrote: > > K_h <KHolmes@sx729.com> wrote: > > > > > > "Frederick Williams" wrote in message > > news:51336CFB.CDE15F2C@btinternet.com... > >> > >> > If we can somehow use sin(x) to represent the number of revolutions, > >> > here's an argument that the cardinality of sin(x) is c^c. > >> > >> It is sets that have a cardinality. What set is sin(x)? > > > > Don put forward a bad argument. The sine function is the set of all points > > (x, sin(x)) where x is any real number. So the cardinality of sin(x) is > > just c, the cardinality of the real line because there are two values in (x, > > sin(x)) and 2*c=c in cardinal arithmetic. > > Agreed, sin(x) is a bad argument for the OP's revolving wheel question. > The physics forum seems to argue that the cardinality of the range > created by feeding the set of rational numbers into sin(x) is c^2.
What physics forum? The cardinality of the range of a function cannot exceed that of its domain. Here I take "function" to mean "single valued function" as is, I think, usual when real functions are discussed. Writing Q for the set of rational numbers and R for the set of real numbers, the function
f: Q > R
defined by
f(x) = sin(x)
is single valued and has a range of cardinality aleph_0.
Here, range(f) = {y : y = f(x) for some x} and is not to be confused with graph or codomain.
 When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

