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\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\I{\mathop{\cal I}\nolimits}
\def\A{\mathop{\cal A}\nolimits}
\def\B{\mathop{\cal B}\nolimits}
\def\N{\mathop{\cal N}\nolimits}
\def\F{\mathop{\cal F}\nolimits}
\def\Po{\mathop{\cal P}\nolimits}
\def\G{\mathop{\cal G}\nolimits}
\def\La{\mathop{\cal L}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\gro{\mathop{\omega}\nolimits}     
\def\grS{\mathop{\Sigma}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\implies{\mathop{\to}\nolimits}     
\def\deduces{\mathop{\vdash}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


\begin{document}
\begin{center} {\bf Problem Set IV}\\
Math 609  \end{center}

The point of this problem set is to give a proof of the Compactness theorem
without any hypotheses on the cardinality of the language.

Let $I$ be a set. A set $\F$ of subsets of $I$ is called a {\em filter} on $I$
if it satisfies the following three conditions.
\begin{description}
\item[i.]  $\emptyset \not\in \F.$
\item[ii.] If $A,$ $B \in \F,$ then $A \cap B \in \F.$
\item[iii.] If $A \in \F$ and $B \supseteq A,$ then $B \in \F.$
\end{description}

\begin{enumerate}
\item Let $A \subseteq I.$ Verify that the set $\F_A = \{ B \subseteq I : B \supseteq A \}$
is a filter. Filters of the form $\F_A,$ $i \in I,$ are called {\em principal.}
\item A subset $A \subseteq I$ is called {\em cofinite} if the complement 
$I \setminus A$ is finite. Verify that the set of all cofinite subsets of an infinite set $I$
is a filter. It is called the Fr\'echet filter.
\item A set $\G$ of subsets of $I$ is said to have the {\em finite intersection
property} (FIP) if the intersection of any finite set of elements from $\G$
is nonempty. Prove that any set of subsets of $I$ with the FIP is contained in
a filter $\F$ on $I$ and that there exists a unique smallest (with respect to inclusion)
filter $\F$ that contains $\G.$ This filter $\F$ is called the filter {\em generated by} $\G.$
\item Let $S$ be an infinite set and $I = \Po_{\gro} (S)$ the set of all finite subsets of $S.$
For $i \in I,$ define $i^* := \{ j \in I : j \supseteq i  \}.$ Show that the set
$\{ i^* : i \in I \}$ has the FIP.
\item Prove that every filter is contained in a filter that is maximal with
respect to inclusion. Maximal filters are called {\em ultrafilters.} 
\item Let $\F$ be an ultrafilter. Prove that for every subset $A \subseteq I,$ $A \in \F$ or 
$I \setminus A \in \F.$ 
\item Show that every principal ultrafilter has the form $\F_{\{ i \}}$ for some element $i \in I.$ 
\item Let $A_1, A_2, \ldots , A_n$ be a finite collection of subsets of $I$ such that the union
$A_1 \cup A_2 \cup \dots \cup A_n = I.$ If $\F$ is an ultrafilter, show that one of the
$A_i$ belongs to $\F.$
\item Prove that an ultrafilter on an inifinite set $I$ that is not principal contains the Fr\'echet filter.
\item Let $\La$ be a first-order language and $\F$ be an ultrafilter on $I.$ For a set
$\{ A_i : i \in I \}$ of $\La$-structures, define the relation $a \equiv_{\F} b$ on the cartesian
product $\prod_{i \in I} A_i$ by 
$$ a \equiv_{\F} b \mbox{ iff } \{ i :  a_i = b_i  \} \in \F.$$
We say that $a$ and $b$ are equal almost everywhere (with respect to $\F$). Prove that the relation
$a \equiv_{\F} b$ is an equivalence relation. The class of $a$ modulo this relation is denoted
$a / \F.$
\item Denote by $\prod_i A_i / \F$ the set of equivalence classes modulo the relation $\equiv_{\F}.$
It is called the {\em ultraproduct} of the $A_i$ with respect to $\F.$ Define an $\La$-structure on
$\prod_i A_i / \F.$
\item Prove \L o\'s's Theorem. For every formula $\grf ( x_0, x_1, \ldots, x_n)$ of $\La$ in the free
variables $x_0, x_1, \ldots, x_n$ and $a_0, a_1, \ldots, a_n \in \prod_i A_i,$
$$\prod_i A_i / \F \models \grf [ a_0 / \F, a_1 / \F, \ldots, a_n / \F] \mbox{ iff }
A_i \models \grf [ (a_0)_i, (a_1)_i, \ldots, (a_n)_i] \mbox{ for almost all } i.$$
\item Use \L o\'s's Theorem (and exercise 4) to prove that every finitely satisfiable set $\grS$
of formulas of $\La$ is satisfiable.
\end{enumerate}
\end{document}