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Systems for which Gödel theorems are provable Anil Mitra, December 2013—December 2013 To have a Gödel theorem be provable for a system the system must have enough structure to be able to describe a statement that refers to itself as an unprovable (Gödel) statement. Arithmetic with multiplication, addition, and first order logic is rich enough; Presburger arithmetic (no multiplication, but multiplication can be simulated by additions) and Euclidean Geometry are not rich enough. Roughly, the two Gödel theorems are (1) Sufficiently rich and consistent systems cannot be complete and (2) The consistency of such systems cannot be proved within the system. Interestingly, if the Gödel statement were false it could be proved and so must be true; therefore, since the statement says it is unprovable it must be unprovable; and adding it as a theorem does get around the theorems because then another Gödel statement can be found. |