Model Theory (Lecture)

I’m following this course by Dr. Vincent Bagayoko on Model Theory at the University of Constance during the 2nd Semester. A recap of the material relevant for the exam follows.

Briefly, Theorems and Defs.

• Def. Filter: (i) $\varnothing \notin \mathcal{F},I\in \mathcal{F}$, (ii) Closed under $\cap$ (iii) Closed under supersets in $I$
  • Principal Filter: If $x\in I$ then $\mathcal{F}$ is generated by $x$, if $x$ is a singleton, then $\mathcal{F}$ is ultra
  • Fréchet Filter: $F=\{A\subseteq I|I\setminus A\text{ is finite}\}$
  • Free Filter: Contains a Fréchet filter
  • Ultrafilter: (i) ${\forall }_{A,C\subseteq I}A\cup B\in \mathcal{F}\Rightarrow A\in \mathcal{F}$ or $B\in \mathcal{F}$ (prime filter on $\cup$) or (ii) is a maximal filer
  • FIP: If the intersection over any finite subcollection of $I$ is non empty
  • Creation of a Filter: If $\mathcal{A}\subset \mathcal{P}(I)$ has the FIP, then there is a filter $\mathcal{F}$ on $I$ with $\mathcal{A}\subseteq \mathcal{F}$
  • Extension of a Filter: $\mathcal{F}\cup \{A\}$ is contained is a filter on $I$.
  • Ultraproducts: ${\prod }_{i\in I}{\mathcal{M}}_i/\mathcal{U}$
    • Łos Theorem: ${\prod }_{i\in I}{\mathcal{M}}_i/\mathcal{U}\models \varphi (m_1,\dots ,m_n)\iff \{i\in I:M_i\models \varphi (m_{1,i},\dots ,m_{n,i})\}\in \mathcal{U}$.
  • Ultrapower: ${\prod }_{i\in I}\mathcal{M}/\mathcal{U}$
• Embedding:(i) constants (ii) functions (iii) relations but only injective for the variables
  • I. Type: $p$ is a $n$-type of $Th(\mathcal{M})$ $\iff$ e.ext. $\mathcal{N}$ of $\mathcal{M}$ s.t. ${\exists }_{b\in N}\mathcal{N}\models p(b)$ $\iff$ ${\forall }_{p_0\subset p}{\exists }_{b\in M}\mathcal{M}\models p_0(b)$
  • II. Type: $t{p}^{\mathcal{M}}(a)$ is the complete type of $a$ in $\mathcal{M}$
  • Elementary Embedding: $\mathcal{M}{\models }_h\varphi \iff \mathcal{N}{\models }_{\tau \circ h}\varphi$
  • Elementary Equivalence, $\equiv$**: $\mathcal{M}\models \varphi \iff \mathcal{N}\models \varphi$
  • Elementary Extension**: $\mathcal{M}\subseteq \mathcal{N}$ s.t. for all $\bar{a}\in {\bigcup }_{n\in \omega }{M}^n$, $t{p}^{\mathcal{M}}(\bar{a})=t{p}^{\mathcal{N}}(\bar{a})$.
  • Substructure: $N\subseteq M$, $\mathcal{N}$ closed and same outputs on everything $\mathcal{M}$ has.
  • Elementary Substructure: $\mathcal{M}\models \varphi \iff \mathcal{N}\models \varphi$ and $\mathcal{N}$ substr. of $\mathcal{M}$
    • E. Emb $\Rightarrow$ E. Eq p. 72
  • Compactness Theorem: Any finite subset of $T$ is consistent $\iff$ $T$ is consistent.
  • Corr. to Comp. If $\Gamma \models \varphi$ then there is a finite ${\Gamma }_0$ s.t. ${\Gamma }_0\models \varphi$.
  • Löwenheim-Skolem Theorem: Inf. $\mathcal{M}$, inf. $k$, then there is an E. Ex. or E. Sub. $|\mathcal{N}|=k$.
  • Definable Choice: $\mathcal{M}$ has def. choice over $A$ if for every form. with param. in $A$ $\varphi (v_0,...,v_n)$ there is a partial func. (not def on all $M^n$) $f:M^n\to M$, $\mathcal{M}\models \varphi (m_0,...,m_{n-1},f(m_0,...,m_{n-1}))$
• Quantifier Elimination: For $\varphi \in \mathcal{L}$ there is $\psi \in \mathcal{L}$ w.q., s.t. $T\models \psi \leftrightarrow \varphi$ and with $FV(\varphi )=FV(\psi )$
  • QE $\Rightarrow$ Model Completeness: For every str. s.t. $\mathcal{M},\mathcal{N}\models T$: $\mathcal{M}\subseteq \mathcal{N}\Rightarrow \mathcal{M}⪯\mathcal{N}$.
  • T.-V. test: $\mathcal{N}⪯\mathcal{M}\iff {\forall }_{\varphi (c,y_1,...,y_n)}{\forall }_{b_1,...,b_n\in N}\mathcal{M}\models {\exists }_x\varphi (x,b_1,...,b_n)\to {\exists }_{a\in N}\mathcal{M}\models \varphi (a,b_1,...,b_n)$.
  • QE $\Rightarrow$ Def. Sets without Quantifiers are all Def. Sets.
  • 1. Criterion for QE, $T$ has QE $\iff$ If $\mathcal{M},\mathcal{N}\models T$, $\mathcal{A}\subseteq \mathcal{M},\mathcal{N}$: for $\bar{a}\in A^n$, $\mathcal{M}\models \varphi (\bar{a})\iff \mathcal{N}\models \varphi (\bar{a})$.
  • 2. Criterion for QE: Just prove you can eliminate $\exists$.
  • O-minimality on $\mathcal{M}$: Every definable subset $X\subseteq M$ is a finite union of intervals and points.
  • O-min $\iff \mathcal{M}\models \varphi (v_0)\leftrightarrow \psi (v_0)$ where $\psi (v_0)\in {\mathcal{L}}_{<}$… true?
  • O- min on $T$: Every model of $T$ has O-min
  • $T$ complete $\Rightarrow$ just one O-min str. needed
  • Use QE for O-min: QE $\wedge$ Sets def. by atomic $\varphi (v_0)$ are finite unions of intervals $\Rightarrow$ O-min.
  • We know: abelian groups, DLO, algebraically closed fields, real closed fields are QE
  • We know: DLO in ${\mathcal{L}}_{>}$, real closed fields are o-min
  • RCF: $\mathbb{F}[i]$ is ACF $\iff$ $\mathbb{F}\equiv \mathbb{R}$
• Categoricity: $T$ is $k$-cat. iff any two models of $T$ of card, $k$ are isomorphic.
• Vaught’s test: if all $\mathcal{M}\models T\Rightarrow |\mathcal{M}|\ge {\aleph }_0$, $k\ge max(|\mathcal{L}|,{\aleph }_0)$ and $T$ is $k$-cat. then $T$ is complete.