[Logic] [Mathematics] [Model Theory] - Introduction To Model Theory (W. Hodges).pdf

Introduction to model theory
Wilfrid Hodges
Queen Mary, University of London
This course is an introduction in two senses. First, it is for people who
haven’t studied model theory before, though I trust most people in the class
will have heard of it. I have tried to make most of the material accessi-
ble to ing from any of the main disciplines where model theory
is used: mathematics, philosophy, computer science, linguistics, cognitive
psychology.
And second, I have tried to start where model theory starts. This course
is an introduction to the basic notions rather than to the most impressive
achievements.
Since we are discussing the starting points, I dip back into history per-
haps more than is usual in introductory courses.
Unfortunately I haven’t yet found time to put plete notes.
What you have here is a provisional schedule of topics, and some historical
and other material that I expect to refer to when we discuss these topics.
1
DAY ONE: Schemas, formulas, models, structures
The texts quoted here illustrate important steps in the early development
of the notion of a model.
From David Hilbert, Foundations of geometry (1899), [6] §9:
Consider a pair of numbers (x, y) from the
eld
[the
eld of
algebraic numbers] as a point and the ratios (u : v : w)ofany
three numbers from
as a line provided u, v are not both zero.
Furthermore, let the existence of the equation
ux + vy + w =0
mean that the point (x, y) lies on the line (u : v : w). Thereby,
as is easy to see, Axioms I, 1-3 and IV are immediately satis
ed.
[Axiom I,1 said ‘For every two points A, B there exists a line a
that contains each of the points A, B’.]
From Oswald and Young, Projective Geometry II (1918) [17]:
This is represented in
g. 19 and may be realized in a model by
cutting out a rectangular strip of paper, giving it a half twist,
and pasting together the two ends.
...
plete the model it would be necessary to bring the two
edges labeled