Godel's theorem is one of the most interesting and revolutionary theorems of mathematics ever produced. It states that any sufficiently complex axiomatic system will, if it is consistent, be incomplete - that is to say there will be statements that can neither be proved nor disproved. Moreover the necessary complexity of the theory is low - any theory that can express elementary arithmetic is complex enough to fall within its scope. Godel's theorem rests on the idea of godel-numbering propositions - encoding any proposition or string of basic symbols as a number in such a way that the syntactic operations of implication and so on can be turned into arithmetic operations that operate on the godel numbers of the propositions. Then using an ingenious diagonal argument , it is possible to construct a statement that refers to itself in such a way as to deny its own provability. This statement, the Godel statement, then clearly cannot be proved or disproved, in a way analogous to the liar paradox - 'this statement cannot be proved' rather than 'this statement is false'.
Many people maintain that this statement is clearly true but unprovable. This is not so as truth must be defined with respect to a model, and the model is not uniquely determined by the axioms. Moreover since the Godel statement cannot be proved, its negation can be consistently added to the set of axioms, and a model produced in which it is be definition false. These models are called non-standard models of arithemtic, The existence of these non-standard models shows that the Godel statement is indeterminate in truth value with respect to the axioms of arithmetic, though it is true in the standard model, the natural numbers.
| Number of referees | 7 |
| Abstract & title | 6.0 |
| Audience | 3.9 |
| Presentation | 5.4 |
| Pacing & timing | 5.6 |
| Questions | 5.7 |
| Overall % | 72.0 |
Referees' comments: Relied heavily on overheads which, for me, were obstructed. Shame. As a non-mathematician, subject matter went over my head. Very good, interesting topic. Has understood the theorem. A bit elementary though.
