模型论是数学的新门类
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“模型论”,其中第一句话是.Model
Theory is the part of mathematics which shows how to apply logic to
the study of structures in pure mathematics. On the one hand it is
the ultimate abstraction; on the other, it has immediate
applications to every-day mathematics. The fundamental tenet of
Model Theory is that mathematical truth, like all truth, is
relative. A statement may be true or false, depending on how and
where it is interpreted. T
阅读之后了就会明日歌曲哦了。
袁萌陈启清2月8日
附件:
Fundamentals of Model Theory
William
Weiss and Cherie D’Mello
Introduction
Model
Theory is the part of mathematics which shows how to apply logic to
the study of structures in pure mathematics. On the one hand it is
the ultimate abstraction; on the other, it has
immediate applications to every-day mathematics. The fundamental
tenet of Model Theory is that mathematical truth, like all truth,
is relative. A statement may be true or false, depending on how and
where it is interpreted. This isn’t necessarily due to mathematics
itself, but is a consequence of the language that
we use
to express mathematical ideas. What at rst seems like a deciency
in our language, can actually be shaped into a powerful tool for
understanding mathematics. This book provides an
introduction to Model Theory which can be used as a text for a
reading course or a summer project at the senior undergraduate or
graduate level. It is also a primer which will give someone a self
contained overview of the subject, before diving into one of the
more encyclopedic standard graduate texts. Any reader who is
familiar with the cardinality of a set and the algebraic closure of
a eld can proceed without worry. Many readers will have some
acquaintance with elementary logic, but this is not absolutely
required, since all necessary concepts from logic are reviewed in
Chapter 0. Chapter 1 gives the motivating examples; it is short and
we recommend that you peruse it rst, before studying the more
technical aspects of Chapter 0. Chapters 2 and 3 are selections of
some of the most important techniques in Model Theory. The
remaining chapters investigate the relationship between Model
Theory and the algebra of the real and complex numbers. Thirty
exercises develop familiarity with the denitions and consolidate
understanding of the main proof techniques. Throughout the book we
present applications which cannot easily be found elsewhere in such
detail. Some are chosen for their value in other areas of
mathematics: Ramsey’s Theorem, the Tarski-Seidenberg Theorem. Some
are chosen for their immediate appeal to every mathematician:
existence of innitesimals for calculus, graph colouring on the
plane. And some, like Hilbert’s Seventeenth Problem, are chosen
because of how amazing it is that logic can play an important role
in the solution of a problem from high school algebra. In each
case, the derivation is shorter than any which tries to avoid
logic. More importantly, the methods of Model Theory display
clearly the structure of the main ideas of the proofs, showing how
theorems of logic combine with theorems from other areas of
mathematics to produce stunning results. The theorems here are all
are more than thirty years old and due in great part to the
cofounders of the subject, Abraham Robinson and Alfred Tarski.
However, we have not attempted to give a history. When we attach a
name to a theorem, it is simply because that is what mathematical
logicians popularly call it. The bibliography contains a number of
texts that were helpful in the preparation of this manuscript. They
could serve as avenues of further study and in addition, they
contain many other references and historical notes. The more recent
titles were added to show the reader where the subject is moving
today. All are worth a look. This book began life as notes for
William Weiss’s graduate course at the University of Toronto. The
notes were revised and expanded by Cherie D’Mello and
2
William
Weiss, based upon suggestions from several graduate students. The
electronic version of this book may be downloaded
and further modied by anyone for the purpose of learning, provided
this paragraph is included in its entirety and so long as no part
of this book is sold for prot.
Contents
Chapter
0. Models, Truth and Satisfaction 4 Formulas,
Sentences, Theories and Axioms 4 Prenex Normal Form
9
Chapter
1. Notation and Examples 11
Chapter
2. Compactness and Elementary Submodels 14 The Compactness Theorem
14 Isomorphisms, elementary equivalence and complete theories 15
The Elementary Chain Theorem 16 The
L¨owenheim-Skolem Theorem 19 The L o´s-Vaught Test 20 Every complex
one-to-one polynomial map is onto 22
Chapter
3. Diagrams and Embeddings 24 Diagram Lemmas 25 Every planar graph
can be four coloured 25 Ramsey’s Theorem 26 The Leibniz
Principle and
innitesimals 27 The Robinson Consistency Theorem 27 The Craig
Interpolation Theorem 31
Chapter
4. Model Completeness 32 Robinson’s Theorem on existentially
complete theories 32 Lindstr¨om’s Test 35 Hilbert’s Nullstellensatz
37
Chapter
5. The Seventeenth Problem 39 Positive denite rational functions
are the sums of squares 39
Chapter
6. Submodel Completeness 45 Elimination of quantiers 45 The
Tarski-Seidenberg Theorem 48
Chapter
7. Model Completions 50 Almost universal theories 52 Saturated models 54 Blum’s Test 55
Bibliography 60
Index
61
3
CHAPTER
0
Models,
Truth and Satisfaction
We will
use the following symbols: • logical symbols: – the
connectives ∧
,∨
, ¬ , →
, ↔ called “and”, “or”, “not”, “implies” and “i”
respectively – the quantiers ∀ ,
∃ called “for all”
and “there exists” – an innite collection of variables indexed by
the natural numbers N v0 ,v1 , v2 , ... – the two parentheses ), (
– the symbol = which is the usual “equal sign” • constant
symbols : often denoted by
the letter c with subscripts • function symbols : often denoted by
the letter F with subscripts; each function symbol is an m-placed
function symbol for some natural number m ≥
1 •
relation symbols : often denoted by the letter R with
subscripts; each relational symbol is an n-placed relation symbol
for some natural number n ≥
1.
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