\section{Structures}

\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}

\definition{}
A \textit{structure} consists of a universe and a set of \textit{symbols}. \par
A structure's symbols give meaning to the objects in its universe.

\vspace{2mm}

Symbols come in three types:
\begin{itemize}
	\item \textit{Constant symbols}, which let us specify specific elements of our universe. \par
		Examples: $0, 1, \frac{1}{2}, \pi$
		\vspace{2mm}

	\item \textit{Function symbols}, which let us navigate between elements of our universe. \par
		Examples: $+, \times, \sin{x}, \sqrt{x}$ \par
		\note{Note that symbols we usually call \say{operators} are functions under this definition. \par
		The only difference between $a + b$ and $+(a, b)$ is notation.}
		\vspace{2mm}

	\item \textit{Relation symbols}, which let us compare elements of our universe. \par
		Examples: $<, >, \leq, \geq$ \par
		\vspace{2mm}
\end{itemize}

The equality check $=$ is \textit{not} a relation symbol. It is included in every structure by default. \par
By definition, $a = b$ is true if and only if $a$ and $b$ are the same element of our universe.


\vspace{3mm}


\example{}
The first structure we'll look at is the following:
$$
	\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
$$

\vspace{2mm}

This is a structure over the universe $\mathbb{Z}$ that provides the following symbols:
\begin{itemize}
	\item Constants: \tab $\{0, 1\}$
	\item Functions: \tab $\{+, -\}$
	\item Relations: \tab $\{<\}$
\end{itemize}

\vspace{2mm}

If we look at our set of constant symbols, we see that 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 this structure offers.

\vspace{2mm}

% NOTE: this is a great example for typesetting.
% The line breaks here are ugly without a centered sentence.
To \say{define} an element of a set, we need to write a sentence that is only true for that element. \par
If we want to define 2 in the structure above,
we could use the following sentence:
\begin{center}
	\say{$2$ is the $x$ that satisfies $[1 + 1 = x]$.} \par
\end{center}
This is a valid definition because $2$ is the \textit{only} element of $\mathbb{Z}$ for which $[1 + 1 = x]$
evaluates to \texttt{true}.


\problem{}
Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$.

\begin{solution}
	The sentences \say{$x$ where $[x + 1 = 0]$} and \say{$x$ where $[0 - 1 = x]$} both work.
\end{solution}

\vfill
\pagebreak







Let us formalize what we found in the previous two problems. \par

\definition{Formulas}
A \textit{formula} in a structure $S$ is a well-formed string
of constants, functions, relations, \par and logical operators.

\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.

\vspace{2mm}

As a quick example, the formula $\psi \coloneqq [\lnot (1 = 1)]$ is always false, \par
and $\varphi(x) \coloneqq [1 + 1 = x]$ evaluates to \texttt{true} only when $x$ is 2.





\definition{Free Variables}
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.

\vspace{2mm}

For example, consider the two formulas from the previous definition, $\psi$ and $\varphi$:
\begin{itemize}
	\item $\psi \coloneqq [\lnot (1 = 1)]$ \par
	There are no free variables in this formula. \par
	In any structure, $\psi$ is always either \texttt{true} or \texttt{false}.

	\vspace{2mm}

	\item $\varphi(x) \coloneqq [1 + 1 = x]$ \par
	This formula has one free variable, labeled $x$. \par
	The value of $\varphi(x)$ depends on the $x$ we're talking about: \par
	$\varphi(72)$ is false, and $\varphi(2)$ is true.
\end{itemize}

\vspace{2mm}

\note{
	This \say{free variable} notation is very similar to the function notation we are used to: \par
	The values of both $\varphi(x) \coloneqq [x > 0]$ and $f(x) = x + 1$ depend on $x$.
}




\definition{Definable Elements}
Let $S$ be a structure over a universe $U$. \par
We say an element $x \in U$ is \textit{definable in $S$} if we can write a formula $\varphi(x)$ that only $x$ satisfies.


\problem{}
Define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$. \par
\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.}

\begin{solution}
	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) \coloneqq [x \times x = 4]$.
\end{solution}


\vfill
\pagebreak






\problem{}
Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$. \par
Why can't you do it?

\begin{solution}
	We could try $\varphi(x) \coloneqq [x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \par
	We have no way to distinguish between negative and positive numbers. \par

	\note{This problem is intentionally hand-wavy. We don't have the tools to write a proper proof.}

	\begin{instructornote}
		Actually, it is. Bonus problem: how? \par
		Do this after understanding quantifiers.
	\end{instructornote}
\end{solution}

\vfill


\problem{}
Consider the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$

\begin{itemize}
	\item Define $2^2$
	\item Define $2^n$ for all positive integers $n$
	\item Define $2^{-n}$ for all positive integers $n$

	\item What other numbers can we define in this structure? \par
	\hint{There is at least one more \say{class} of numbers we can define.}
\end{itemize}


\begin{solution}
	As far as I've seen, we can define any $2^{\nicefrac{a}{b}}$ for $a, b \in \mathbb{Z}$. \par
	For example, $\phi(x) \coloneqq [2 = x \div (1 \div x)]$ defines $\sqrt{2}$.
\end{solution}

\vfill
\pagebreak