Lesson 3

As an example, consider the operations over whole numbers \(\mathbb{N}= \{0, 1, 2, … \}\). Note that, throughout this section, we treat \(0\) as a whole number. We can express sentences about whole numbers in first-order logic by adopting the vocabulary \(\Sigma_{\mathbb{N}}= (\Phi_{\mathbb{N}}, \Pi_{\mathbb{N}}, r_{\mathbb{N}})\), defined below. \[\begin{align*}\Phi_{\mathbb{N}}&= \{0, \sigma, +, \times, \uparrow\}, \\ \Pi_{\mathbb{N}}&= \{ =, < \},\\ r_{\mathbb{N}}&= \{ 0 \to 0, \sigma \to 1, + \to 2, \uparrow \to 2, = \to 2, < \to 2 \},\end{align*}\] where \(\sigma\) is the successor function and \(\uparrow\) is the exponentiation.

Problem. Here is an expression in number theory: \[\times\left(+(x, \sigma(\sigma(0))), \; \sigma\left(\uparrow\left(x, \sigma(\sigma(0))\right)\right)\right).\] Rewrite this formula in usual math style (i.e. something like \(x \cdot (x + 1)\)).