## Mathematical Definability

### BibTeX

@MISC{Slaman_mathematicaldefinability,

author = {Theodore A. Slaman},

title = {Mathematical Definability},

year = {}

}

### OpenURL

### Abstract

Introduction One might fairly say that the mathematical analysis of definability began in 1931, with the appearance of Godel's Incompleteness Theorem [Godel, 1931]. Godel showed that for sufficiently strong formal systems T , there exist undecidable statements ' such that there is no proof of ' or of :' within T . This theorem pointed to an intrinsic incompleteness within the formal notion of proof. The method of computation by algorithm is more general than that of verification by formal proof. It too was shown to be incomplete, but it took some time to develop the technical apparatus needed to state this incompleteness correctly. Kleene [1987], in his biographical memoir of Godel, recalls this development and we summarize some of his remarks. Kleene describes the intuitive notion of an algorithm as follows. An algorithm is a procedure described in advance such that, whenev