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\flushleft \underline{Homework 11 -- MATH 2510 Spring 2019}\\
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Consider the following assignment of values for symbols: 
\begin{center}
\begin{tabular}{|l|l|}
\hline 
Symbol in the theory & Assigned numerical value \\
\hline
$S$ & 2 \\
$0$ & 3 \\
$\cdot$ & 4 \\
$=$ & 5 \\
$($ & 6 \\
$)$ & 7 \\
$+$ & 8 \\
\hline
\end{tabular}
\end{center}

Here is a (portion of a) proof from HW6 \#2b:
\[ \begin{array}{ll}
(1) S0 \cdot SS0 = (S0 \cdot S0) + S0 \\
(2) S0 \cdot S0 = S0 \\
(3) S0 \cdot SS0 = S0 + S0
\end{array}\]

\begin{enumerate}[1.]
\item Find the G\"{o}del number of each of the three lines in the proof.
\item Find the super G\"{o}del number of the proof above.
\item Consider the set $I=\{0,1,2,3\}$. List all subsets of $I$.
\item Is the set $\mathcal{F}=\bigg\{ \{0\}, \{0,1\}, \{0,2\}, \{0,1,2,3\} \bigg\}$ a filter on $I$? If so, explain why it satisfies all axioms of a filter. If not, explain which axioms of a filter fail for this $\mathcal{F}$. 
\item Is the set $\mathcal{F}=\bigg\{ \{1\}, \{0,1\}, \{1,2\}, \{1,3\}, \{0,1,2\}, \{0,1,3\}, \{1,2,3\}, \{0, 1,2,3\} \bigg\}$
a filter? If so, explain why it is satisfies all axioms of a filter. If not, explain which axioms of a filter fail for this $\mathcal{F}$.
\end{enumerate}
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