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\begin{document}
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\begin{center}

  \huge{The Department of Mathematics}\\[0.1\baselineskip]
  \Large{2019--20--B term}\\[0.2\baselineskip]

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\begin{description}
  \item[Course Name]
    Introduction to Set Theory

  \item[Course Number]
    \LRE{201‭.1‭.0171}

  \item[Course web page]\mbox{}\\
    \url{https://math.bgu.ac.il//en/teaching/spring2020/courses/set-theory}

  \item[Lecturer]
    Prof. Menachem Kojman,
    \nolinkurl{<kojman@bgu.ac.il>},
    Office 111

\item[Office Hours]
  \url{https://math.bgu.ac.il/en/teaching/hours}
\end{description}

\section*{Abstract}




\section*{Requirements and grading\footnote{Information may change during the first two weeks of the term. Please consult the webpage for updates}}






\section*{Course topics}

\begin{enumerate}
\item{} Partially ordered sets. Chains and antichains. Examples. Erdos--Szekeres' theorem or a similar theorem. The construction of a poset over the quotient space of a quasi-ordered set.
\item{} Comparison of sets. The definition of cardinality as as an equivalence class over equinumerousity. The Cantor-Bernstein theorem. Cantor's theorem on the cardinality of the power-set.
\item{} Countable sets. The square of the natural numbers. Finite sequences over a countable set. Construction of the ordered set of rational numbers. Uniqueness of the rational ordering.
\item{} Ramsey's theorem. Applications.
\item{} The construction of the ordered real line as a quotient over Cauchy sequences of rationals.
\item{} Konig's lemma on countably infinite trees with finite levels. Applications. A countable graph is k-colorable iff every finite subgraph of it is k-colorable.
\item{} Well ordering. Isomorphisms between well-ordered sets. The axiom of choice formulated as the well-ordering principle. Example. Applications. An arbitrary graph is k--colorable iff every finite subgraph is k-colorable.
\item{} Zorn's lemma. Applications. Existence of a basis in a vector space. Existence of a spanning tree in an arbitrary graph.
\item{} Discussion of the axioms of set theory and the need for them. Russel's paradox. Ordinals.
\item{} Transfinite induction and recursion. Applications. Construction of a subset of the plane with exactly 2 point in every line.
\item{} Infinite cardinals as initial ordinals. Basic cardinal arithmetic. Cardinalities of well known sets. Continuous real functions, all real runctions, the automorphisms of the real field (with and without order).
\end{enumerate}

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