Mirna's class  notes
 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.
 Examples
 Groups (note the choice of language)
 Pure sets
 Rings and fields
 Vector spaces (discuss choice between two languages)
 Peano arithmetic
 Graphs
 DLO

The Compactness Theorem. A theory $\Gamma$ is satisfiable if and only if it is satisfiable.
 Countable models and
\(I(T,\aleph_{0})\)
 $\omega$categoricity: examples and nonexamples
 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 $I(T,\aleph_{0})=n$.
 RyllNardzewski 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 \(\aleph_{0}<I(T,\aleph_{0})<2^{\aleph_{0}}\).
 $\omega$categoricity: examples and nonexamples
 Miscellany
 Biembeddability versus isomorphism: examples
 Elementary version
 Uncountable spectrum
 $I(T,\kappa)$
 Morley’s Theorem.
 Shelah’s Main Gap and its philosophy