- #1

David Carroll

- 181

- 13

As you know, the identity function involves taking two axes perpendicular to each other and then depicting a straight line at a 45

^{o}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

^{.5}u 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.