From bd01da032991f9671557ef64e23ca684fa6c995a Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sat, 9 Jul 2022 13:44:47 +0200
Subject: Introduction corrected

---
 sections/introduction.tex | 55 +++++++++++++++++++++++++++--------------------
 1 file changed, 32 insertions(+), 23 deletions(-)

(limited to 'sections/introduction.tex')

diff --git a/sections/introduction.tex b/sections/introduction.tex
index aedc345..433f3e9 100644
--- a/sections/introduction.tex
+++ b/sections/introduction.tex
@@ -1,33 +1,42 @@
 \documentclass[../lic_malinka.tex]{subfiles}
 
 \begin{document}
-  Model theory is a field of mathematics that classify and construct
-  structures with particular properties. It desribes classical mathematical
-  objects in a broader context, abstract their properties and study
-  connections between simingly unrelated structures. Roland Fraïssé was
-  French logician who established many important notions in contemporary
-  model theory. He was one of the first to utilize back-and-forth argument,
-  a fundamental model theoretical method in construction of 
-  elementary equivalent structures. The Ehrenfeuht-Fraïssé games is a 
-  concept that proved useful in classical logic, model theory, but also
-  finite model theory (which is a filed of theoretical informatics rather
-  than mathematics). 
+  Model theory is a field of mathematics that classifies and constructs
+  structures with particular properties (particularly those expressible
+  in first order logic). It describes classical mathematical
+  objects in a broader context, abstracts their properties and studies
+  connections between seemingly unrelated structures. 
+  This work studies limits of Fraïssé classes with additional combinatorial
+  and categorical properties. Fraïssé classes are frequently used in model 
+  theory,
+  both as a source of examples and to analyse particular ``generic'' structures.
 
-  This work study limits of Fraïssé classes with additional combinatorial
-  and categorical properties. The key theorem \ref{theorem:generic_aut_general}
-  says that a Fraïssé class with canonical amalgamation and weak Hrushovsky
-  property has a generic automorphism. This result was known before,
-  for example [DODAC GDZIE TO BYLO...]. However, we show a new way to construct
-  a generic automorphism by extending the structures of the class by an
-  automorphism and considering limit of such extended Fraïssé class. We achieve
-  this by using the Banach-Mazur games, a well known objects of general topology
-  which prove useful in study of comeager sets.
+  The notion of Fraïssé class and its limit is
+  due to the French logician Roland Fraïssé. He also introduced the
+  back-and-forth argument, a fundamental model theoretical method in 
+  construction of elementarily equivalent structures, upon which
+  Ehrenfeucht-Fraïssé games are based.
 
-  The prototype structure of the paper is the random graph (also known as the 
+  The prototypical example for this paper is the random graph 
+  \ref{definition:random_graph} (also known as the 
   Rado graph), the Fraïssé limit of the class of finite undirected graphs.
   It serves as a useful example, gives an intuition of the Fraïssé limits,
-  weak Hrushovsky property and free amalgamation.
+  weak Hrushovski property and free amalgamation. Perhaps most importantly,
+  the random graph has a so-called generic automorphism (DODAĆ DEFINICJĘ
+  GENERYCZNEGO AUTOMORFIZMU I ZLINKOWAĆ TUTAJ), which was first proved
+  by Truss in \cite{truss_gen_aut}, where he also introduced the term. 
 
-   
+   The key theorem \ref{theorem:generic_aut_general}
+  says that a Fraïssé class with canonical amalgamation and weak Hrushovski
+  property has a generic automorphism. The fact that such an automorphism
+  exists in this case follows from the classical results of Ivanov \cite{ivanov_1999}
+  and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}, we show a new way to construct
+  a generic automorphism by expanding the structures of the class by a (total)
+  automorphism and considering limit of such extended Fraïssé class. We achieve
+  this by using the Banach-Mazur games, a well known method in the descriptive
+  set theory, which proves useful in the study of comeagre sets.
 
+  Finally, we show how this construction of the generic automorphism can be
+  used to deduce some properties of generic automorphisms 
+  (see \ref{proposition:fixed_points}, (COŚ JESZCE)).
 \end{document}
-- 
cgit v1.2.3

