# In the history of mathematics, has there ever been a mistake?

I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But it seems totally possible that this could have happened back in the Enlightenment.

Feel free to interpret this how you wish!

In 1933, Kurt Gödel showed that the class called $$[∃∗∀2∃∗,all,(0)]\lbrack\exists^*\forall^2\exists^*, {\mathrm{all}}, (0)\rbrack$$ was decidable. These are the formulas that begin with $$∃a∃b…∃m∀n∀p∃q…∃z\exists a\exists b\ldots \exists m\forall n\forall p\exists q\ldots\exists z$$, with exactly two $$∀\forall$$ quantifiers, with no intervening $$∃\exists$$s. These formulas may contain arbitrary relations amongst the variables, but no functions or constants, and no equality symbol. Gödel showed that there is a method which takes any formula in this form and decides whether it is satisfiable. (If there are three $$∀\forall$$s in a row, or an $$∃\exists$$ between the $$∀\forall$$s, there is no such method.)