Question from a non-mathematician to any HN mathematician...
Has any work been done on the continuum theory by viewing different infinite sets as dimensions (like arrays) of infinity rather than sizes of infinity?
Perhaps it's a superfluous distinction, but it seems that an infinite set of integers can be viewed as a 1-dimensional infinity. The set of all real numbers between 0 & 1 (or any 2 consecutive integers) could equivalently be viewed as a 1-D infinity. However, the set of real numbers between 0 and 2 (or any 3 consecutive integers) would be a 2D infinity, and by extension the set of all real numbers would be an infinite dimensional infinity.
Has any work been done on the continuum theory by viewing different infinite sets as dimensions (like arrays) of infinity rather than sizes of infinity?
Perhaps it's a superfluous distinction, but it seems that an infinite set of integers can be viewed as a 1-dimensional infinity. The set of all real numbers between 0 & 1 (or any 2 consecutive integers) could equivalently be viewed as a 1-D infinity. However, the set of real numbers between 0 and 2 (or any 3 consecutive integers) would be a 2D infinity, and by extension the set of all real numbers would be an infinite dimensional infinity.