As we know, it has been proven that the set of all of points with real components in a cube is equipotent with the set of real numbers. Now, consider the set of all of points built with all of the natural numbers of three axes. Is this set equipotent with the set of natural number? Any help will be appreciated.
Suppose an infinite-dimensional space with basis vectors of Ík (kϵN). Now, every arbitrary cross product of Ím˄ Ín (as it is defined in 3-D space) will uniquely result in a certain Íp. Can we be assured that every Íp is indeed coincident with one of the Ík? In other word, can we do all of the possible cross products assuring that the set of all of the resulted vectors is the same (neglecting the sequence of members) as Ík? If yes, how it can be shown?
Cartesian product of sets: .
Sorry, I am to blame. I will try to describe my problem just in the related field. Suppose we built the set of all pairs of (m, n), m & n ϵ N and m≠n. Then, we want to build the set of all of (m, n, l), l ϵ N, l≠m,n. Now, can we be assured that when m and n are varied all over the N, l remains in N too? In other word, can we take the sets of m, n, and l the same (as N)?