I believe this is set theory, but I don't really know set theory, so...
An autonym is a word that refers to a property that the word itself happens to possess. For example "noun" is an autonym because "noun" is a noun. "Word" is an autonym. "Pentasyllabic" is too. "Poop" is not, unless you can poop words. (I can't.)
So. Is "autonym" an autonym? If you can't prove "yes" or "no" exclusively, does that make it a restatement of some axiom of set theory?