An Elucidation of Goedel's Incompleteness Theorems

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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
Stream: Knowledge and Technology
Presentation Type: Virtual Presentation in English
Paper: A paper has not yet been submitted.

William Robert Catton

PhD Student, Department of Physics, University of Otago
Dunedin, New Zealand

I was born in Christchurch, New Zealand, and spent my childhood in New Zealand and Canada, with one year of high school in Leeds, U.K. While in Leeds I applied to Trinity College at the University of Cambridge, where I ended up obtaining a Masters Degree in Physics. I then returned to New Zealand to enroll in the Energy Studies program at the University of Otago, where I am studying toward a PhD in the numerical simulation of Heat Pump Drying Systems. Aside from thermodynamics, computational modeling, and the implementation of sustainable energy solutions, my chief interest is in the (evolutionary) origins of consciousness -- the main purpose being, of course, to enjoy these precious conscious moments!

Ref: T08P0406