**Paradoxes are extremely interesting as they seem to challenge our fundamental ideas on how the world works. People have been looking at paradoxes for thousands of years and there is still discussion on some of the oldest paradoxes people have recorded. What makes them so interesting is that most of them are true in a mathematical/logical way but they don’t make any sense intuitively. Here are some examples:**

Happiness or a ham sandwich?

Happiness or a ham sandwich?

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.

Smullyan (1), p. 219

**Proof that there exists a unicorn**

I

wish to prove to you that there exists a unicorn. To do this it

obviously suffices to prove the (possibly) stronger statement that

there exists an existing unicorn. (By an existing unicorn I of course

mean one that exists.) Surely if there exists an existing unicorn, then

there must exist a unicorn. So all I have to do is prove that an

existing unicorn exists. Well, there are exactly two possibilities:

(1) An existing unicorn exists.

(2) An existing unicorn does not exist.

Possibility

(2) is clearly contradictory: How could an existing unicorn not exist?

Just as it is true that a blue unicorn is necessarily blue, an existing

unicorn must necessarily be existing.

**"Interesting" and "uninteresting" numbers **

The

question arises: Are there any uninteresting numbers? We can prove that

there are none by the following simple steps. If there are dull

numbers, then we can divide all numbers into two sets – interesting and

dull. In the set of dull numbers there will be only one number that is

the smallest. Since it is the smallest uninteresting number it becomes,

*ipso facto *, an interesting number. We must therefore remove

it from the dull set and place it in the other. But now there will be

another smallest uninteresting number. Repeating this process will make

any dull number interesting.

**The ship of Theseus**

The ship wherein Theseus and the youth of Athens

returned had thirty oars, and was preserved by the Athenians down even

to the time of Demetrius Phalereus, for they took away the old planks

as they decayed, putting in new and stronger timber in their place,

insomuch that this ship became a standing example among the

philosophers, for the logical question of things that grow; one side

holding that the ship remained the same, and the other contending that

it was not the same.

Plutarch, *Vita Thesei*, 22-23

**Proving that
2 = 1**

Here

is the version offered by Augustus De Morgan: Let *x* = 1.

Then *x*² = *x*. So *x*² – 1 = *x* -1. Dividing

both sides by *x* -1, we conclude that *x* + 1 = 1; that

is, since *x* = 1, 2 = 1.

Quine,

p.5

Assume

that

*a*

= *b*. (1)

Multiplying

both sides by *a*,

*a*²

= *ab*. (2)

Subtracting

*b*² from both sides,

*a*²

– *b*² = *ab* – *b*² . (3)

Factorizing

both sides,

(*a*

+ *b*)(*a* – *b*) = *b*(*a* – *b*). (4)

Dividing

both sides by (*a* – *b*),

*a*

+ *b* = *b*. (5)

If

now we take *a* = *b* = 1, we conclude that 2 = 1. Or

we can subtract *b* from both sides and conclude that *a*,

which can be taken as *any* number, must be equal to zero.

Or we can substitute *b* for *a* and conclude that any

number is double itself. Our result can thus be interpreted in a

number of ways, all equally ridiculous.

Northrop,

p. 85

The paradox

arises from a disguised breach of the arithmetical prohibition on

division by zero, occurring at (5): since a = b, dividing

both sides by (a – b) is dividing by zero, which renders

the equation meaningless. As Northrop goes on to show, the same

trick can be used to prove, e.g., that any two unequal numbers are

equal, or that all positive whole numbers are equal.

Of course paradoxes are everywhere, you just need to look carefully

http://www.paradoxes.co.uk/

http://en.wikipedia.org/wiki/List_of_paradoxes