I think I might have found a proof that the irrational positive real numbers (including transcendentals) are uncountable that involves the use of a very simple picture: the identity function graph. As you know, the identity function involves taking two axes perpendicular to each other and then depicting a straight line at a 45o to either axis and meeting at the crossing point of said axes. Now, according to the Pythagorean Theorem, for every range unit of length "u" on either axis, there is a corresponding length 2.5u on the actual graph. Now that length "u" can be viewed as a domain on either the x- or y-axis. Only the identity function has this feature such that the domain is necessarily the same as the range, since the two are completely interchangeable. Thus, we take a domain from the y- axis which transforms into a range on the graph line itself, and from there it re-transforms into a domain on the x-axis (or vice versa), getting identity. The one problem is that any domain "u" will undergo a distortion as it translates into a 2.5 range on the graph line itself. Now if all positive real numbers were countable, then some numbers on the graph line itself could not possibly appear on either x-axis or y-axis since the range is different from the domain (by a factor of 2.5), which would reveal that some numbers are not identical with their own selves. But since this is clearly a contradiction, then the set of all positive real numbers is not countable.