Yes, yes, we all appreciate the people that read. We are awed at the mere sight of them and deep down inside, we all envy them… even if we don’t admit it.

Indeed all books may look like an arch-enemy when you don’t have time for them. And your personal library may become your very own personal Hell with piles of threatening books and rivers of words flowing menacingly.

Well, sometimes, Hell is a library. Luckily, only in fiction. Carlo Frabetti’s Hell Book is a very brief journey in Dante’s Inferno, Greek mythology and mathematics. Don’t be scared though.

It’s made of short stories that are paradoxical, the least to say. If stripped down of the logical and mathematical ideas, the book has the same structure as Dante’s Inferno. The main character is dragged through the nine circles of Hell, having to answer a mathematical question at the exit of each.

Does it sound like a piece of cake? Would you like to see if you could get out of Hell? Consider this situation: suppose you’d have access to all the books in the world and you were asked to make a register of all the non- self- referential books. Could that book really be **complete**?

The correct answer is no. The Register of Non- Self- Referential Books ought to be in this register. The register in itself is indeed non- self- referential book. But, if included, Register of Non- Self- Referential Books would become a self- referential book; thus the paradox.

How many of you still think you would be able to escape Hell?

Comments

## Jesse

Actually, the correct answer is yes. As you said, there are two possible ways that a complete registry can be made: one is by including every self-referential book (without including the registry, which, unless referenced in a registry entry, would not be self-referential); the other is by including every self-referential book, including the registry.

This is not a paradox.

## Le Poissonchat

You make a compelling argument. Yet, if the registry does not include itself, it is incomplete(because it is a non-self-referential book and thus should be included in the Registry of Non-Self-Referential books) but if you include it in itself, it becomes a self-referential book- thus contradicting its purpose. In the latter case the book would be complete indeed, but it would be incorrect.

## Jesse

Ah, I understand. Criticism withdrawn.

## monkeylizard

I think you’ve confused “self-referential” with “non self-referential”. It would indeed be possible to make a complete registry of all books that reference themselves, since in either case, the register would be accurate.

However, the paradox occurs when you ask for a register of all *non* self-referential books, since if the register does not include itself, it becomes a non self-referential book, in which case, it must include itself, which causes it to be a self-referential book, and disqualified from inclusion, etc.

See Russell’s paradox: http://en.wikipedia.org/wiki/Russell%27s_paradox

## Jesse

I did indeed confuse non self-referential with self-referential. Thanks to you and Le Poissonchat for the replies.

## Le Hobbit

Who wouldn’t have confused the damn two?

## Andre

The Classic of Russell’s Paradox.