Edits

Advanced/Definable Sets/parts/1 structures.tex

@@ -1,39 +1,36 @@
 \section{Structures}
 
-\definition{}<def:language>
-A \textit{language} is a set of meaningless objects. Here are a few examples:
+\definition{}
+A \textit{universe} is a set of meaningless objects. Here are a few examples:
 \begin{itemize}
 	\item $\{a, b, ..., z\}$
 	\item $\{0, 1\}$
 	\item $\mathbb{Z}$, $\mathbb{R}$, etc.
 \end{itemize}
 
-Every language comes with the equality check $=$, which checks if two elements are the same.
-
-
 \definition{}
-A \textit{structure} over a language $\mathcal{L}$ consists of a set of symbols. \par
-The purpose of a structure is to give a language meaning.
+A \textit{structure} consists of a universe $U$ and set of symbols. \par
+A structure's symbols give meaning to the objects in its universe.
 
 \vspace{2mm}
 
 Symbols generally come in three types:
 \begin{itemize}
-	\item Constant symbols, which let us specify specific elements of our language. \par
+	\item Constant symbols, which let us specify specific elements of our universe. \par
 		Examples: $0, 1, \frac{1}{2}, \pi$
 		\vspace{2mm}
 
-	\item Function symbols, which let us navigate between elements of our language. \par
+	\item Function symbols, which let us navigate between elements of our universe. \par
 		Examples: $+, \times, \sin{x}, \sqrt{x}$
 		\vspace{2mm}
 
-	\item Relation symbols, which let us compare elements of our language. \par
+	\item Relation symbols, which let us compare elements of our universe. \par
 		Examples: $<, >, \leq, \geq$ \par
-		The symbol $=$ is \textbf{not} a relation. Why? \hint{See \ref{def:language}}
 		\vspace{2mm}
-
 \end{itemize}
 
+The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default.
+
 \vspace{3mm}
 
 
@@ -47,7 +44,7 @@ $$
 
 \vspace{2mm}
 
-This is a structure over $\mathbb{Z}$ with the following symbols:
+This is a structure with the universe $\mathbb{Z}$ that contains the following symbols:
 \begin{itemize}
 	\item Constants: \tab $\{0, 1\}$
 	\item Functions: \tab $\{+, -\}$
@@ -86,10 +83,17 @@ A \textit{formula} in a structure $S$ is a well-formed string of constants, func
 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.
 
+\vspace{2mm}
+
+A formula can contain one or more \textit{free variables.} These are denoted $\varphi{(a, b, ...)}$. \par
+Formulas with free variables let us define \say{properties} that certain objects have. \par
+
+For example, $x$ is a free variable in the formula $\varphi(x) = x > 0$. \par
+$\varphi(3)$ is true and $\varphi(-3)$ is false.
 
 \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.
+Say $S$ is a with a universe $U$. \par
+We say an element $e \in U$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies.
 
 
 \problem{}
@@ -104,10 +108,10 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 
 
 \problem{}
-Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$?
+Try to 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 $-2$. \\
+	This isn't possible. 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}