As strange as it may seem, it is possible to give a precise-looking verbal definition of an integer that, in fact, is not a definition at all. The following was devised by an English librarian, G. G. Berry, and reported by Bertrand Russell. Explain how it leads to a contradiction. Let n be “the smallest integer not describable in fewer than 12 English words.” (Note that the total number of strings consisting of 11 or fewer English words is finite.)

As strange as it may seem, it is possible to give a precise-looking verbal definition of an integer that, in fact, is not a definition at all. The following was devised by an English librarian, G. G. Berry, and reported by Bertrand Russell. Explain how it leads to a contradiction. Let n be “the smallest integer not describable in fewer than 12 English words”. (Note that the total number of strings consisting of 11 or fewer English words is finite).

Given information:

The letter n denotes the smallest integer not describable in fewer than 12 English words.

Concept used:

Note that, there is a self-reference in the description of n and there is finite number of strings consisting of 11 or fewer words.

Calculation:

That means, finitely many phrases can be formed with fewer than 12 words.

So, there must be finite number of strings that describe integers.

Thus, there are finitely many integers that can be defined by phrases of fewer than 12 words.

But, there are infinitely many integers. It means that, there are some integers that cannot be defined by phrases of fewer than 12 words