h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

"This is division by zero, which is illegal in mathematics, and indeed, this is why it's illegal. "

Division by zero is by no means illegal in maths, the result is infinity (represented by the sleeping 8). It's in _computers_ division by zero is 'illegal', and computers can't calculate 2+2, so don't ask it to compute (or by any means try to store the answer to) a division by zero.

Most 'Two Plus Two Equal Five' paradoxes simply proves that X*infinity = Y*infinity, Y+infinity = X+infinity ad infinitum (only exception of course; 0*infinity = 1 for a reasonably sized 0).

Whether infinity itself should be considered a paradox is another matter. Some might argue that Infinity and Eternity are the only true paradoxes, and that if you break down all other paradox-wannabes you will end up proving Infinity.

VOGON
** "To Infinity and Beyond!!"
(- What does that mean?
- Ya know, beyond infinity.
- What's infinity?
- Well, way out there is infinity. I think it's like a planet or something.)
-Buzz

I find your descussions about infinity logical enough when applied to the Zenon Paradox about the turtle (or the simplified version of it which is represented in the guide entry "Paradox" which brought forth the infinite number of converstation forums brimfull with stimulating thoughts, amongst which your contribution is noticeable and stands apart.)
As regards this Zeno Paradox, The Guide Entry Researcher proposes to explain it by stating that what the paradox confuses is the infinite divisibility of space vs the infinite extendability thereof i.e. if infinitly divisible time is juxtaposed upon the infinitly devisible space,clarity of resolution will speak for itself- and that I take leave to doubt.In your notation it is expresible as the sleeping 8, that proxies for space, reclining upon another somniferous eight,standing for time.
With all due respect,the proper analogy for it is akin to prisoners huddled together on a double-decker bed, locked away from the redeeming rays of daylight,oblivious of the sun of clarity.

My proposition to resolve the Zeno paradox is rooted in the idea that though we have to walk the infinite amount of legs of the journey to reach the opposite door, each following leg of our journey will be twice shorter the the former one. i.e. to reach the door I need to walk all the distance to it; the acomplisment whereof ,on the other hand, requires me to reach half the distance first,than the quarter of that distance,eighth, sixthteenth part of it and so forth ad infinitum.
But at the moment when the task to approach the door seems absolutely daunting and unacomplishable , I am reminded of the fact that what I have in hand is no more than the receding geometrical progression: 1+1/2+1/4+1/8... And though the number of such additives in the progression be infinite, our math schoolbooks tell us that their sum is strictly finite and determinable.
I have no reason to disbelieve those textbooks and ,therefore, regain my shaken confidence that planes will travel through the sky, trains will conect distant places far and near, and my door will not remain shut to the knockings of friends for long.
But, however,I find it always jarring in me that finity can be engendered by infinity.For that reason it is always a miracle that we can use integral and difirential calculus so safely and that it stands us in such a good stead.

"To see the world in the grain of sand
And heaven in the wild flower
To see infinity in one hand
And eternity in one hour"