Definition 9.2 (ACF with characteristic). For $n\ge 1$, let ${\chi }_n$ be the sentence

$$\underset{n\text{ times}}{\underbrace{1+\cdots +1}}=0.$$

Set

$$\begin{array}{llllll}\hfill {\mathrm{ACF}}_0& =\mathrm{ACF}\cup \{\neg {\chi }_n:n\in \mathbb{N}\}& \hfill & & \hfill & \\ \hfill {\mathrm{ACF}}_p& =\mathrm{ACF}\cup \{{\chi }_p\}& \hfill & \text{(for }p\text{ a prime)}& \hfill & & \hfill & \end{array}$$