1. We begin by recalling languages, theories, and structures.
• Languages are tuples $((f_{i})_{i\in I},(R_{j})_{j\in J},(c_{k})_{k\in K})$ with associated arities for the function and relation symbols.
• An $L$-structure is a tuple $M=(M,(f_{i}^{M})_{i\in I},(R_{j}^{M})_{j\in J},(c_{k}^{M})_{k\in K})$ where each $f_{i}^{M}$ is an operation on $M$ of the right arity, each $R_{j}^{M}$ is a relation on $M$ of the right arity, and each $c_{k}^{M}$ is an element of $M$.
• We build $L$-formulas and $L$-sentences, and $L$-theories are simply sets of $L$-sentences.
• Given an $L$-structure $M$, and $L$-formula $\varphi(x)$, and an $x$-tuple $a\subset M$, we define ‘satisfaction’ $M\models\varphi(a)$, by recursion on the complexity of $\varphi$.
• We generally assume theories to be consistent: to have a model.
2. Examples
• Groups (note the choice of language)
• Pure sets
• Rings and fields
• Vector spaces (discuss choice between two languages)
• Peano arithmetic
• Graphs
• DLO
3. The Compactness Theorem. A theory $\Gamma$ is satisfiable if and only if it is satisfiable.

4. Countable models and $I(T,\aleph_{0})$
• $\omega$-categoricity: examples and non-examples
• Pure set
• Vector spaces
• Algebraically closed fields
• Random graph
• DLO
• A technical tool:
• Omitting Types Theorem.
• Vaught’s “Not $2$” Theorem. Let $T$ be a complete theory in a countable language. Then $I(T,\aleph_{0})\neq2$. Moreover, for each $n\in\mathbb{N}\setminus{2}$ there exists such a $T$ with $I(T,\aleph_{0})=n$.
• Ryll-Nardzewski Theorem. Let $T$ be a complete theory with infinite models in a countable language. Then $T$ is $\omega$-categorical iff and only if $T$ has finitely many complete $n$-types, for each $n\in\mathbb{N}$.
• Vaught’s Conjecture. Let $T$ be a complete theory in a countable language. Then we do not have $\aleph_{0}<I(T,\aleph_{0})<2^{\aleph_{0}}$.
5. Miscellany
• Bi-embeddability versus isomorphism: examples
• Elementary version
6. Uncountable spectrum
• $I(T,\kappa)$
• Morley’s Theorem.
• Shelah’s Main Gap and its philosophy