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