Added

Advanced/Definable Sets/parts/1 structures.tex

@@ -84,7 +84,6 @@ Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, +, -, <\} \Bigr)$? \par
 \pagebreak
 
 Let us formalize what we found in the previous two problems. \par
-\say{Definable elements} are one of the two most important ideas in this handout.
 
 \definition{}
 A \textit{formula} in a structure $S$ is a well-formed string of constants, functions, and relations. \par
@@ -99,15 +98,6 @@ For the sake of time, I will not provide a formal definition. It isn't particula
 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.
 
-\problem{}
-Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$?
-
-\begin{solution}
-	No. We could try $[x \text{ where } x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \\
-	We have no way to distinguish between negative and positive numbers.
-\end{solution}
-
-\vfill
 
 \problem{}
 Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par
@@ -119,6 +109,18 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 
 \vfill
 
+
+\problem{}
+Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$?
+
+\begin{solution}
+	No. We could try $[x \text{ where } x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \\
+	We have no way to distinguish between negative and positive numbers.
+\end{solution}
+
+\vfill
+
+
 \problem{}
 What is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?