 In this way, syntactical properties, relations and operations are reflected in arithmetic: for example, \(\textit{neg}(x)\) is the arithmetical function that sends the Gödel number of a formula to the Gödel number of its negation; in other words, \(\textit{neg}(\ulcorner A\urcorner) = (\ulcorner \neg A\urcorner)\); similarly, \(\impl(x, y)\) is the function which maps the Gödel numbers of a pair of formulas to the Gödel number of the implication of the formulas: \(\impl(\ulcorner A\urcorner ,\ulcorner B\urcorner) = \ulcorner A \rightarrow B\urcorner\); and so on. There is an arithmetical formula, call it \(\textit{Fmla}(x)\), which is true of \(\boldsymbol{n}\) iff \(\boldsymbol{n}\) is a Gödel number of a well-formed formula of the system. There is also an arithmetical formula \(M(x,y,z)\) which is true exactly if one has a valid application of the rule of inference modus ponens for some formulas \(A\) and \(B\) with \(x = \ulcorner A\urcorner\), \(y = \ulcorner A \rightarrow B\urcorner\) and \(z = \ulcorner B\urcorner\); etc. In this way, all the syntactic properties and operations can be simulated at the level of numbers, and moreover they are strongly representable in all theories which contain Q.