The arithmetic of cuts in models of arithmetic (Q2856636)

If \(M\) is a model of arithmetic, say \(M \models\mathrm{PA}\), then \(I \subseteq M\) is a cut in \(M\) if \(I\) contains \(0\) and is closed under the successor function. The set of all cuts in \(M\) has an ``obvious'' additive structure, with \(I+J\) defined as the supremum of \(i+j\) over all \(i \in I\), \(j \in J\). However, addition of cuts could be defined just as naturally by letting \(I \oplus J\) equal the infimum of \(i+j\) over all \(i>I\), \(j>J\), and it turns out that the two definitions do not always coincide.NEWLINENEWLINEThe paper is mostly devoted to an investigation of the operations \(I+J\) and \(I \oplus J\). The author shows that both these operations satisfy most of the usual properties of addition, and characterizes the cases when \(I+J = I \oplus J\). To achieve the latter aim, the author introduces the notion of the \textit{derivative} of a cut. The derivative of \(I\), \(\partial I\), is a canonically defined cut contained in \(I\) and closed under addition (readers familiar with Solovay's technique of shortening cuts will recognize \(\partial I\) as the result of shortening \(I\) in the usual way to make it closed under \(+\)). The operation \(\partial\) has some formal properties in common with the derivative familiar from elementary calculus. The author shows that \(I+J = I \oplus J\) whenever \(\partial I\) and \(\partial J\) differ; and if \(\partial I = \partial J\), then \(I+J \neq I \oplus J\) exactly if there exists some element \(a \in M\) such that \(I+J\) equals \(a - \partial I\) and \(I \oplus J\) equals \(a + \partial I\).NEWLINENEWLINEThe paper concludes with some preliminary results concerning the joint additive and multiplicative structure of cuts. Here, it is most convenient to consider cuts in the fraction field \(Q\) of \(M\) rather than in \(M\) itself, since in this setting it is easier to give natural definitions of the logarithm and the exponential of a cut. The logarithm makes it possible to reduce some questions about the multiplicative structure to questions about the additive structure. Moreover, the cases when the two multiplicative analogues of \(+\) and \(\oplus\) coincide can be characterized in terms of the second derivative, where \(\partial^2 I\) equals \(\exp(\partial(\log I))\).