Grelling-Nelson Word Paradox

The following paradox was formulated by Kurt Grelling and Leonard Nelson in 1908. It is derived from Russell's Paradox.

A homogeneous (or autological) word is a word which describes itself. The word "short" is homogeneous because it is a short word. Likewise, polysyllabic is homogeneous because it is itself a polysyllabic word.

A heterogeneous (or heterological) word is a word which does not describe itself. Long and monosyllabic are both heterogeneous.

It would therefore seem logical to suggest that all words can either be described as homogeneous, or heterogeneous, and can thus be put in to one of these two sets. Right?

WRONG - There is a problem with this idea. Can you work it out?

It feels as though there might be something paradoxical about the answer to a problem being the problem but anyway...

What happens when one tries to put heterogeneous in to one of the two sets?

Is heterogeneous heterogeneous?

On the face of it, it doesn't seem to describe itself, so let us try putting it in to the set of words which do not describe themselves. This is the heterogeneous set. But if heterogeneous is heterogeneous, then it does describe itself and must therefore be homogeneous. But if it is homogeneous, it does not describe itself...in which case it must be heterogeneous, and so ad infinitum.

It would seem, therefore, that the word heterogeneous cannot be put in to either set , as it bounces straight out of whichever set it is put in to. If it is homogeneous it is heterogeneous and if it is heterogeneous it is homogeneous; clearly a contradiction.

The poem The Long and the Short of it was inspired by this paradox.