 Both notions of representability—strong and weak—must be clearly distinguished from mere definability (in the standard sense of the word). A set \(S\) is definable in the language of arithmetic if there is a formula \(A(x)\) in the language such that \(A(\underline{n})\) is true in the standard structure of natural numbers (the intended interpretation) if and only if \(\boldsymbol{n} \in S.\) There are many sets which can be defined in the language of arithmetic but not (even weakly) represented in any \(F\), such as the set of consistent formulas, the set of sentences unprovable in the system \(F\), or the set of Diophantine equations with no solutions (see below).