“…no statement which refers to a ‘reality’ transcending the limits of all possible experience can possibly have any literal signiﬁcance; from which it must follow that… those who have striven to describe such a reality have all been devoted to the produc*on of nonsense.” Alfred Jules Ayer, “The Elimina*on of Metaphysics”
Gödel’s First Incompleteness Theorem Any eﬀec5vely generated theory capable of expressing elementary arithme5c cannot be both consistent and complete. In par5cular, for any consistent, eﬀec5vely generated formal theory that proves certain basic arithme5c truths, there is an arithme5cal statement that is true, but not provable in the theory.
In other words…. An arithme*c system, for instance a ﬁnite set of axioms, cannot be BOTH consistent and complete.
where…. Consistent –> contains no logical/mathema*cal contradic*ons Complete –> describes all possible logical/ mathema*cal statements.
In other words…. …there is an arithme*c statement that is true, but not provable by the theory. Finite lists of axioms cannot describe a system where all statements are shown to be true/ false.