2011/05/02 by Stephen Boyce, Boyce, Stephen
Computer Science · #Advanced Database Systems and Queries #Logic, programming, and type systems #Semantic Web and Ontologies
paper · pdf · doi:10.48550/arxiv.1105.0321
The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class of PA theorems is semantically defined by a class sign of PA itself, (E x2) Pf(x2, x1), in the following sense: with b' the PA numeral for the number b, (E x2) Pf(x2, b') is true under the standard interpretation if and only if b is the Godel number of a PA theorem. From this however it is easily established, by a modification of Godel's proof, that the class of PA theorems, and hence the standard interpretation of PA itself, is not well defined after all.