Paradoxes... Part One.

Don't know what a paradox is?

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Here are a couple of paradoxes to think about. The paradoxes are obtained from:

• Falletta: The Paradoxicon
• Sainsbury: Paradoxes, 2nd Edition
• Smullyan: What Is The Name of This Book?
• Morris: The Ivan Morris Puzzle Book
  

I know, it's weird to put sources first, but I'm too damned lazy to type them out afterwards.

God is not all-powerful as he cannot build a wall he cannot jump .

A Paradoxical notice:
Please ignore this notice.

Ham Sandwich > Eternal Happiness.
Which is better, eternal happiness or a ham sandwich? It would appear that eternal happiness is better, but this is really not so! After all, nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore a ham sandwich is better than eternal happiness.

The Simplest Liar Paradox:
This Sentence is False.
Is that sentence true or false? If it is false then it is true, and if it is true then it is false...

Tortoise > Achilles
Zeno's second paradox of motion, of Achilles and the tortoise, is probably the best known of his four paradoxes of motion. In this problem, the fleet Greek warrior runs a race against a slow-moving tortoise. Assume Achilles runs at ten times the speed of the tortoise (1 meter per second to 0.1 meter per second). The tortoise is given a 100-meter handicap in a race that is 1,000 meters. By the time Achilles reaches the tortoise's starting point T₀, the tortoise will have moved on to point T₁. Soon, Achilles will reach point T₁, but by then the tortoise would have moved on to T₂, and so on, ad infinitum. Every time Achilles reaches a point where the tortoise has just been, the tortoise has moved on a bit. Although the distances between the two runners will diminish rapidly, Achilles can never catch up with the tortoise, or so it would seem.

Sorites Paradox of the Heap:
Suppose you have a heap of sand. If you take away one grain of sand, what remains is still a heap: removing a single grain cannot turn a heap into something that is not a heap. If two collections of grains of sand differ in number by just one grain, then both or neither are heaps. This apparently obvious and uncontroversial supposition appears to lead to the paradoxical conclusion that all collections of grains of sand, even one-membered collections, are heaps.

Proving that 2 = 1

Here is the version offered by Augustus De Morgan: Let x = 1. Then x² = x. So x² - 1 = x -1. Dividing both sides by x -1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1.