# Indexing problems

Can you make sense of this paradoxical question?

## Problem

Consider a very big book, UltraRef, which refers in its index to every book which does not refer to itself in its index.

Does UltraRef refer to itself in its index? Be clear as to your explanation.

Did you know ... ?

Paradoxes are a fascinating feature of more advanced considerations of mathematical logic. For a century mathematicians tried to put mathematics on a totally sure logical foundation before realising that this could never be completely achieved.

Paradoxes are a fascinating feature of more advanced considerations of mathematical logic. For a century mathematicians tried to put mathematics on a totally sure logical foundation before realising that this could never be completely achieved.

## Student Solutions

Constructing a really clear argument to this problem can be difficult!

Put simply:

Suppose that exactly one of the following statements is true:

A: Ultra Ref does refer to itself in its index

B: Ultra-Ref does not refer to itself in its index

Look at each in turn:

IF (Ultra-Ref does refer to itself in its index) THEN (Ultra-Ref does not refer to itself in its index)

IF (Ultra-Ref does not refer to itself in its index) THEN (Ultra-Ref does refer to itself in its index)

In both cases, our supposition is contradicted and Ultra-Ref is therefore a logical impossibility.