Report I can't or I won't say in New Logic/Math Puzzles Posted October 18, 2012 · Edited October 18, 2012 by mmiguel mmiguel, would you say there are different cardinal numbers for one edge, 12 edges, faces, solids? BTW just to clarify, I mean by cube to denote a solid, as per EH's reasoning. Any other views? i was interpreting the OP to mean solid. i did a little research, and it seems that by the standard mathematical definition of cardinality, it seems that just because an infinite set A is a superset of an infinite set B, this does not mean that |A| > |B|. the concept that the superset operator translates into a "greater-than" inequality when going from sets to the cardinalities of those sets is true for finite sets, as is the bijective mapping view of cardinality that mathematicians have chosen to stick to when generalizing from finite sets to infinite sets. under this generalization, the two concepts are no longer equivalent. this is something i may have read a long time ago, but did not recall when providing my answer. http://en.wikipedia....iki/Cardinality "Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite setS that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it." so what this means is that a solid cube has the same cardinality as an edge of the cube because a bijective mapping can be formed between the two sets. i wonder, is there a standard definition for the infinite-set generalization of the subset/superset interpretation of cardinality from the finite case? perhaps such a definition was deemed to be not useful... thinking over this, it is all so very strange: the cube contains all the points that the edge contains, and then some.... yet each point in the cube can be paired with a point in the edge, with no points in the edge being left unpaired... if these are both true, than this must be a contradiction (mustn't it?). if one is false, then why is it false? i think it must be a contradiction if both are true because suppose we set about pairing points in the cube to the edge. let's say we restrict ourselves to first pair all the points in the edge of the cube before pairing any other points of the cube. since the cube contains the edge, we could pair each point of the edge from the cube set perspective, to the same point in the edge set perspective. each point in the edge gets paired with itself. thus there are no longer any points in the edge that are unpaired. but there are points not on the edge, which still are part of the cube which are not paired to any points within the edge, and cannot be, since all the points in the edge set are already paired (with themselves from the cube set perspective). hence no bijective mapping can be formed if restrict ourselves to pairing the points within the edge first in the cube set perspective.... perhaps someone can enlighten me?