 The undecidable sentences provided by Gödel’s proofs are (if written out) extremely complicated formulas with no intuitive significance, construed only for the purposes of the incompleteness proofs. The question then arises whether there are any simple and natural mathematical statements which are likewise undecidable in chosen basic theories, e.g., in PA. There are now various specific statements with clear mathematical content which are known to be undecidable in some standard theories (though, just how natural even these are has been disputed; see Feferman 1989b). Some well known, natural examples are listed below, beginning with some quite natural mathematical statements which are independent of PA, and proceeding to more and more powerful theories. Sometimes such results are called variants of Gödel’s theorem, or their proofs of independence alternative proofs of Gödel’s theorem, but this is misleading: interesting as they may be, they don’t have the generality of Gödel’s theorems proper, but only provide statements independent of a particular theory.