Julia Robinson proved that the theory of fields is undecidable by showing that the natural numbers form a subset of the rationals definable by a first-order formula in the language of fields. The construction of this formula involved an ingenious application of number theory. For an accessible account, see the article by Flath and Wagon:
- Stan Wagon and Dan Flath, How to pick out the integers in the rationals: an application of number theory to logic, Amer. Math. Monthly 98 (1991), no. 9, 812–823. MR 1132996 (93b:03076)
