Now when you combine the Completeness and Incompleteness Theorems, you can get some really remarkable results. If you work with the axioms of number theory, call them N (which include many of the above axioms F along with axioms for < and axioms for mathematical induction), for example, we know by the Incompleteness Theorem that there is a statement X such that neither X nor (not X) is provable. Hence, by the Completeness Theorem, there is a model of N in which X is true and a model of N in which X is false.

It follows that there are mathematical universes which look and act very much like the regular natural numbers, but do in fact have some subtle differences. One of the most fascinating results I’ve seen is that there is a model of number theory which “thinks” (in a precise sense) that the axioms N are inconsistent, even though they are not (roughly, the “proof” of an inconsistency that it “sees” is infinitely long, and so is not a real proof).

— Godel’s Completeness Theorem

— Joe Mileti

2012.12.02 Sunday ACHK