Godel’s Second Incompleteness Theorem in Limerick

— Manoj Gopalakrishnan

 

once i took a class in logic
where with cunning diabolic
i learnt to prove
that i couldn’t prove
that i couldn’t prove
that one equals two
without making Peano melancholic




Author’s note: This limerick is an accurate statement of Godel’s second incompleteness theorem, which considers the question of the consistency of arithmetic.

If we ever find a proof for a false statement, say 1 = 2, then our system of arithmetic is inconsistent. Therefore, to prove that arithmetic is consistent is to prove that there is no proof for 1 = 2. Clearly an inconsistent system, where every statement can be proved, also allows a proof that there is no proof showing 1 = 2. Godel’s remarkable result showed that the converse is also true: if a system of arithmetic proves its own consistency, then it must be inconsistent.

The “Peano” in the limerick refers to Giuseppe Peano, an Italian mathematician, after whom the axioms of our system of arithmetic are named. Surely, melancholy would overtake him if he were to learn that arithmetic was inconsistent!

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