An Elucidation of Goedel's Incompleteness Theorems
Kurt Goedel's stunning incompleteness theorems fundamentally altered the perceived relationship between mathematics (the study of pattern) and its incarnation as a systematized interaction between mathematician, pencil and paper (mathematics as technology). With Goedel's discovery that every consistent formal system for arithmetic must be incomplete, it became apparent that no final such systematization (no final "technology of mathematics") could ever be achieved. This short essay uses several novel devices to introduce the key ideas behind Goedel's famous incompleteness results: (1) a pictorial representation of the relationship between the formal system whose incompleteness Goedel first proved, and its image in the natural numbers; (2) a schematic depiction of the diagonalisation argument, by which he produced a self-referential arithmetic formula asserting its own unprovability; (3) a gradual paraphrasing of the "English translation" of Goedel's undecideable proposition, clarifying this diagonalisation argument; and (4) a monologue in which the protagonist finally arrives at a conclusion analogous to Goedel's second incompleteness theorem. Repeated reference to the diagram, in particular, allows the relevant ideas to be developed from scratch, using direct language and with a minimum of terminology.
Keywords: Goedel, Incompleteness, Formal System, Mathematics, Arithmetic, Number
William Robert Catton
PhD Student, Department of Physics, University of Otago