A mathematical explanation of Russell’s paradox

If we have a cube with the length of the sides being unit 1, then what is the length of the diagonal of that cube?

The answer is that we have to use Pythagoras’ theorem twice.
The line from D to B in the diagram below completes the triangle DCB, which has a right angle at C.

Thus, by Pythagoras’ Theorem

|DB|² = |DC|² + |CB|² = 1² + 1² = 2

 and thus

|DB| = √2

Now consider the triangle ADB

Triangle ABD is also a right triangle, this time with the right angle at D. Hence, using Pythagoras’ Theorem again

|AB|² = |AD|² + |DB|² = 1² + 2 = 3

and thus

|AB| = √3

So the length of a diagonal of a side is exactly √2 units, and the length of the diagonal of the cube is exactly √3 units.

Both of these are irrational numbers, but not of the same kind, since neither of them can be expressed in terms that include the other. If we simply consider both of them as contradictions, then we can consider √2 as an ambiguity (which can also be called a contradiction of the first degree), and √3 as a paradox (which can also be called a contradiction of the second degree). Then, there aren’t any higher degrees of contradictions, since contradiction turns into consistency beyond paradox (ie, every higher number can be expressed in terms including either √2 or √3), or the other way around within and outside of the parenthesis.

This relation between these irrational numbers is actually the turning point for conceptualization (ie, Russell’s paradox) expressed mathematically. What it says is that the middle on a diagonal is ambiguous, and that the middle of a diagonal of a cube is paradoxically contradictory. The difference between them being that the former is “neither nor”, whereas the latter is “both and” of a contradiction. They are simply the back side of what we evaluate. As such, they, and thus the relation between them, explain why we can’t find what we want to find, and thus are unwanted. This may be the reason why no one has found this relation before I did.

Another contribution to understanding of conceptualization http://menvall.wordpress.com/