129 lines
3.8 KiB
TeX
129 lines
3.8 KiB
TeX
\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 $U$ 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 universe. \par
|
|
Examples: $0, 1, \frac{1}{2}, \pi$
|
|
\vspace{2mm}
|
|
|
|
\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 universe. \par
|
|
Examples: $<, >, \leq, \geq$ \par
|
|
\vspace{2mm}
|
|
\end{itemize}
|
|
|
|
The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default.
|
|
|
|
\vspace{3mm}
|
|
|
|
|
|
\example{}
|
|
\def\structgeneric{\ensuremath{}}
|
|
|
|
The first structure we'll look at is the following:
|
|
$$
|
|
\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
|
|
$$
|
|
|
|
\vspace{2mm}
|
|
|
|
This is a structure with the universe $\mathbb{Z}$ that contains the following symbols:
|
|
\begin{itemize}
|
|
\item Constants: \tab $\{0, 1\}$
|
|
\item Functions: \tab $\{+, -\}$
|
|
\item Relations: \tab $\{<\}$
|
|
\end{itemize}
|
|
|
|
\vspace{2mm}
|
|
|
|
If you look at our set of constant symbols, you'll 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 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.
|
|
|
|
\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 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{}
|
|
Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par
|
|
\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
|
|
|
|
\begin{solution}
|
|
$-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}
|
|
Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$.
|
|
|
|
\begin{solution}
|
|
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}
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}
|
|
What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
|
|
|
|
\begin{solution}
|
|
With the tools we have so far, we can only define powers of two, positive and negative.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak |