math: compactness

Notes/compactness_theorem.md

@@ -0,0 +1,26 @@
+---
+title: Compactness Theorem
+date: 2024-12-15T16:09:38-05:00
+references: []
+tags: [In_Progress]
+---
+
+# Statement and Proofs
+
+{{< env type="theorem" name="Compactness Thoerem; Gödel 1930 & Maltsev 1936" >}}
+
+Let $L$ be a first-order language. If an $L$-theory $T$ is finitely-satisfiable, then $T$ is satisfiable. In fact, $T$ admits a model $M\models T$ with $|M|\leq\max\l\\{|L|,\aleph_0\r\\}$.{{< /env >}}
+
+{{< env type="proof" hide="false" name="from Completeness" >}}
+
+Every $L$-theory is *syntactically-compact*, in the sense that $T$ is consistent iff every $T$ is finitely-consistent.
+> Indeed, if $\phi$ witnesses inconsistency of $T$, then there are finite subtheories $\Delta_0,\Delta_1\subseteq T$ such that $\Delta_0\proves\phi$ and $\Delta_1\proves\lnot\phi$. Their union $\Delta\coloneqq\Delta_0\cup\Delta_1$ is then an inconsistent finite subtheory of $T$, a contradiction.
+
+The {{< link file="completeness_theorem.md" display="Completeness Theorem" type="proved_by" >}} then applies, since if $T$ is finitely-satisfiable, it is finitely-consistent, and hence consistent by the above.<span style="float:right;">$\blacksquare$</span>{{< /env >}}<div class="space"></div>
+
+{{< env type="proof" hide="false" name="with Ultraproducts" >}}
+
+Suppose w.l.o.g. that $T$ is infinite and let $\mc{D}$ be the collection of all finite-subtheories of $T$. For each $\Delta\in\mc{D}$, let $M_\Delta\models\Delta$, and let $X_\Delta\subseteq\mc{D}$ be the subcollection of all finite-subtheories of $T$ containing $\Delta$. Since $X_{\Delta_1}\cap X_{\Delta_2}=X_{\Delta_1\cup\Delta_2}$ for any two $\Delta_i\in\mc{D}$, the collection $F\coloneqq\l\\{X\subseteq\mc{D}\st X\supseteq X_\Delta\textrm{ for some }\Delta\in\mc{D}\r\\}$ is a filter, which extends to an ultrafilter $U\supseteq F$.
+<br>
+&emsp;&emsp;We claim that the {{< link file="ultraproduct.md" display="ultraproduct" type="references" >}} $M\coloneqq\prod_{\Delta\in\mc{D}}M_\Delta/U$ models $T$. Indeed, for any $\phi\in T$, we have $M_\Delta\models\phi$ for all $\Delta\in X_{\l\\{\phi\r\\}}$, and thus $X_{\l\\{\phi\r\\}}\subseteq\l\\{\Delta\in\mc{D}\st M_\Delta\models\phi\r\\}$. Since $X_{\l\\{\phi\r\\}}\in U$, we have $\l\\{\Delta\in\mc{D}\st M_\Delta\models\phi\r\\}\in U$, and hence $M\models\phi$ by {{< link file="ultraproduct.md" display="Łoś’s Theorem" type="proved_by" secID="los_theorem" secDisplay="Łoś’s Theorem" >}}.<span style="float:right;">$\blacksquare$</span>
+{{< /env >}}

Notes/theory.md

@@ -0,0 +1,18 @@
+---
+title: Theory
+date: 2024-12-15T16:05:04-05:00
+references: []
+tags: [In_Progress]
+---
+
+# Motivation & Definition
+
+{{< env type="definition" >}}
+
+Let $L$ be a first-order language. An *$L$-theory* $T$ is a collection of $L$-sentences. A *deductively-closed* theory is a theory that is closed under provability, i.e., $\phi\in T$ whenever $T\proves\phi$.{{< /env >}}
+
+## Complete theories
+
+## Consistent and satisfiable theories
+
+# The Category of Theories

Notes/type.md

@@ -7,10 +7,53 @@ tags: [In_Progress]
 
 # Motivation & Definition
 
-Let $M$ be a model of a first-order theory $T$. What are the possible first-order properties satisfied elements (or tuples) in $M$?
+Let $M$ be a model of a first-order {{< link file="theory.md" display="theory" type="references" >}} $T$. What are the possible first-order properties satisfied elements (or tuples) in $M$?
 
 {{< env type="definition" >}}
 
-Let $n\geq0$. An *$n$-type* is a set $p(\bar{v})$ of $L$-formulas such that the $L_\bar{v}$-theory $p(\bar{v})\cup\Th M$ is satisfiable.{{< /env >}}
+Let $n\geq0$. An *$n$-type* is a set $p(\bar{x})$ of $L$-formulas such that the $L$-theory $\ex\bar{x}p(\bar{x})\cup\Th M$ is satisfiable.{{< /env >}}
 
+It will be useful to also consider types with *parameters* over some subset $A\subseteq M$, which is defined by considering the $L_A$-structure $M$; that is, an *$n$-type over $A$* is a set $p(\bar{x})$ of $L_A$-formulas such that the $L_A$-theory $\ex\bar{x}p(\bar{x})\cup\Th_AM$ is satisfiable.
+<br>
+&emsp;&emsp;Every type $\bar{a}\in M^n$ defines the type $\tp^M(\bar{a}/A)\coloneqq\l\\{\phi\in L_A\st M\models\phi(\bar{a})\r\\}$ over $A$. An $n$-type $p(\bar{x})$ of $M$ over $A$ is:
+* *complete* if either $\phi\in p$ or $\lnot\phi\in p$ for all $L_A$-formulas $\phi$. We write $S^M_n(A)$ for the set of all complete $n$-types over $A$;
+* *realized by $\bar{a}\in M^n$* if $p\subseteq\tp^M(\bar{a}/A)$, in which case we write $M\models p(\bar{a})$, and *omitted by $M$* if no $\bar{a}\in M$ realizes $p$;
+* *isolated by* some satisfiable $L_A$ formula $\psi(\bar{x})$ if $\Th_AM\models\fa\bar{x}(\psi(\bar{x})\rightarrow\phi(\bar{x}))$ for all $\phi\in p$.
 
+It will also be convenient to consider the (complete) theory $T\coloneqq\Th_AM$ and talk about $n$-types of $T$ instead of $n$-types of $M$ over $A$; in this case, we write $S_n(T)$ for $S_n^M(A)$.
+
+# Realizing and Omitting Types
+
+## Realizing types in elementary extensions
+
+{{< env type="proposition" >}}
+
+Every $n$-type $p(\bar{x})$ in $M$ over $A$ can be realized in some elementary extension $N\elemextend M$.{{< /env >}}
+
+{{< env type="proof" hide="false" >}}
+
+It suffices to show that the $L_M$-theory $\ex\bar{x}p(\bar{x})\cup\Diag_\textrm{el}M$ is satisfiable; by the {{< link file="compactness_theorem.md" display="Compactness Theorem" type="proved_by" >}}, it further suffices to show that the $L_M$-sentence $\Delta\coloneqq\ex\bar{x}\phi(\bar{x},\bar{a})\land\psi(\bar{a},\bar{b})$, where $\phi(\bar{x},\bar{a})\in p(\bar{x})$, $M\models\psi(\bar{a},\bar{b})$, $\bar{a}\in A$, and $\bar{b}\in A^c$, is satisfiable.
+<br>
+&emsp;&emsp;To this end, let $N_0\models\ex\bar{x}p(\bar{x})\cup\Th_AM$. Since $\ex\bar{z}\psi(\bar{a},\bar{z})\in\Th_AM$, we have $N_0\models\ex\bar{x}\phi(\bar{x},\bar{a})\land\ex\bar{z}\psi(\bar{a},\bar{z})$. Interpreting the constants $\bar{b}$ as witnesses to $\ex\bar{z}\psi(\bar{a},\bar{z})$ makes $N_0\models\Delta$, as desired.<span style="float:right;">$\blacksquare$</span>{{< /env >}}
+
+<div class="space"></div>
+
+**Remark.** In particular, this shows that a type $p(\bar{x})$ is complete iff $p=\tp^N(\bar{a}/A)$ for some elementary extension $N\elemextend M$. The converse follows since $S^M_n(A)=S^N_n(A)$ for any elementary extension $N\elemextend M$. If $p$ is complete, let $N\elemextend M$ realize $p$, i.e. $p\subseteq\tp^N(\bar{a}/A)$ for some $\bar{a}\in N^n$, from which equality follows by completeness of $p$.
+
+## The Omitting Types Theorem
+
+# Types of Models
+
+## Homogeneous models
+
+{{< env type="lemma" >}}
+
+If $\bar{a}_1,\bar{a}_2\in M^n$ realize the same types of $M$ over $A$, that is, if $\tp^M(\bar{a}_1/A)=\tp^M(\bar{a}_2/A)$, then there is an elementary extension $N\elemextend M$ and an automorphism $h\in\Aut_AM$ sending $\bar{a}_1\mapsto\bar{a}_2$.{{< /env >}}
+
+{{< env type="proof" hide="false" >}}
+
+{{< /env >}}
+
+## Saturated models
+
+## Prime and atomic models

Notes/ultraproduct.md

@@ -0,0 +1,6 @@
+---
+title: Ultraproduct
+date: 2024-12-15T16:27:53-05:00
+references: []
+tags: [In_Progress]
+---