This is a structure over $\mathbb{Z}$ with the following symbols:
\begin{itemize}
\item$\mathcal{C}=\{0, 1\}$\tab\note{(constants)}
\item$\mathcal{F}=\{+, -\}$\tab\note{(functions)}
\item$\mathcal{R}=\{<\}$\tab\note{(relations)}
\end{itemize}
\vspace{2mm}
Let's look at $\mathcal{C}$, our set of constant symbols. The only integers we can directly refer to in this structure are 0 and 1. If we want any others, we must define them using the tools the structure offers.
\vspace{1mm}
Say we want the number 2. We could use the function $+$ to define it: $2\coloneqq[x \text{ where }1+1= x]$\par
We would write this as $2\coloneqq[x \text{ where }+(1, 1)= x]$ in proper \say{functional} notation.
\problem{}
Can we define $-1$ in $\Bigl(\mathbb{Z} ~\big|~ \{0, 1, +, -, <\}\Bigr)$? If so, how?
\vfill
\problem{}
Can we define $-1$ in $\Bigl(\mathbb{Z} ~\big|~ \{0, +, -, <\}\Bigr)$? \par
\hint{In this problem, $1$ has been removed from the set of constant symbols.}
\vfill
\pagebreak
Let us formalize what we found in the previous two problems. \par
\definition{}
A \textit{formula} in a structure $S$ is a well-formed string of constants, functions, and relations. \par
\vspace{2mm}
You already know what a \say{well-formed} string is: $1+1$ is fine, $\sqrt{+}$ is nonsense. \par
For the sake of time, I will not provide a formal definition. It isn't particularly interesting.
\definition{Definable Elements}
Say $S$ is a structure over a language $\mathcal{L}$. \par
We say an element $e$ of $\mathcal{L}$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies.
