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 êk (kϵN). Now, every arbitrary cross product of êm˄ ên (as it is defined in 3-D space) will uniquely result in a certain êp. Can we be assured that every êp is indeed coincident with one of the êk? 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 êk? If yes, how it can be shown?
Is this a new, unrelated problem? If so, you should make a new thread in the "Advanced Algebra" subforum. I don't see how this relates to the original problem about the cardinality of the set of 3D points with natural coordinates. I am not sure what the cross product in an infinite-dimensional space is. The original problem did not speak about vector spaces at all, and by the "product" in post #2 I meant the 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)?