**Unit 1: Model Theory**
*Model theory is a generalization of combinatorics and abstract algebra
in which mathematical structures are described by axioms, and their
structure is examined through homomorphisms and definable sets.
Roughly, we start with some “language,” giving the functions, relations,
and constants that are relevant. In this language, we can “define”
certain properties, and can also check the truth of statements about
whether these properties are satisfied.

We will first define the general concepts of “structure” and “morphism,” which give rise to the language as the list of things that must be “preserved” by the morphism: in a graph, for instance, a morphism is a function that preserves adjacency. The power of this approach is clarified by three major results: the back-and-forth technique to construct isomorphisms, the Löwenheim-Skolem Theorem to construct non-isomorphic structures sharing many properties with an original, and the Compactness Theorem, which allows us to determine some “extra” properties preserved by isomorphism but not definable in the language.

Central questions to keep in mind throughout this unit: What properties can be defined? What properties cannot? When is it possible to specify a particular mathematical object up to isomorphism, and what does this specification look like?*

**Unit 1 Time Advisory**

This unit should take approximately 30 hours to complete.

☐ Subunit 1.1: 5 hours

☐ Sub-subunit 1.1.1: 2 hours

☐ Sub-subunit 1.1.2: 3 hours

☐ Subunit 1.2: 8 hours

☐ Sub-subunit 1.2.1: 2 hours

☐ Sub-subunit 1.2.2: 3 hours

☐ Sub-subunit 1.2.3: 3 hours

☐ Subunit 1.3: 9 hours

☐ Sub-subunit 1.3.1: 3 hours

☐ Sub-subunit 1.3.2: 3 hours

☐ Sub-subunit 1.3.3: 3 hours

☐ Subunit 1.4: 8 hours

☐ Sub-subunit 1.4.1: 2 hours

☐ Sub-subunit 1.4.2: 3 hours

☐ Sub-subunit 1.4.3: 3 hours

**Unit1 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:
- Define the syntax and semantics of first-order logic.
- Distinguish syntactic and semantic features.
- Define morphisms in a general context.
- Define elementarity.
- Use the back-and-forth method to construct isomorphisms.
- Use the Löwenheim-Skolem Theorem to establish non-elementarity.
- Use compactness to establish non-elementarity.

**1.1 Structures and Morphisms**
**1.1.1 Structures and Isomorphism**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 1.1”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
1.1” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 1.1 in its entirety. As you
read, remember the key rule to reading mathematical definitions:
make up examples as you go. Can you make up first-order structures
in different languages? With the same domain? With the same
language but important differences? Use these to build examples of
the later definitions. Also, try to build “near-miss” non-examples:
things that satisfy all but one part of a definition, or that
otherwise nearly satisfy the definition, but not quite. Once you
have done this, solve Exercise 1.1.14. When you are finished, you
can check your answers against the Saylor Foundation’s [“Solutions
for Exercises in Unit 1 – Exercise
1.1.14”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.1.1.pdf)
(PDF).
This reading should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.1.2 Examples**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 1.2”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
1.2” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 1.2 in its entirety. If you are
not familiar with fields and groups, you have two options at this
point: First, you can ignore the terms, and think of a *field* as
being like the rational numbers with addition and multiplication,
and of a *group* as being like the integers under addition or the
invertible 2x2 matrices under matrix multiplication (the latter is
the better example, if you know it). The second option is to look
in, briefly, on [MA231: Abstract Algebra
I](http://www.saylor.org/courses/ma231/), where you can find
definitions. (These are not critical to this course but may make
you feel a bit more comfortable.) In any case, think on these
questions: What are the important properties of each kind of
structure that we might want to be able to express or to require a
homomorphism to preserve? Make a list of important properties for
each kind of structure, and keep the list for the next subunit.
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.2 First-Order Formulas**
**1.2.1 The Syntax**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Chapter 2 Introduction and Section 2.1”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Chapter 2 Introduction and Section
2.1” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read from the beginning of Chapter 2 through
the end of section 2.1. Be careful in reading proofs. Even with a
proof that we know is correct, the right way to read a proof is by
believing, for as long as possible, that it is wrong. Much of this
section is the near-obvious bookkeeping system; that is, the
clerical methods by which we can keep track of necessary details.
Very little will be surprising. However, a careful – even
skeptical – reading of the proofs gives a good idea of the level of
formal detail that is often necessary in logic, even if it is
unnecessary in algebra or analysis. After you have read the
formula, prove the statement given as Exercise 2.1.10. When you are
finished, you can check your answers against the Saylor Foundation’s
[“Solutions for Exercises in Unit 1 – Exercise
2.1.10”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.2.1.pdf)
(PDF).
This reading should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.2.2 The Semantics**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 2.2”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
2.2” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 2.2 in its entirety, solving
Exercise 2.2.4 when you come to it. Then go back to the list of
properties you made in subunit 1.1.2. Write a formula, where
possible, to express each of these properties. Remember, in doing
so, to use the definitions *strictly*: the variables may range only
over elements of the domain (and not, for instance, over other
formulas); only finitely many applications of the rules in the
inductive definition can be made (so that, for instance, one can
have only a finite conjunction). In some cases, it will be
impossible. Two very important points arise here, which will form a
good part of the rest of Unit 1: (a) Many important properties can
be expressed by formulas in the natural signature, and (b) Some
important properties cannot. When you are finished, you can check
your answers against the Saylor Foundation’s [“Solutions for
Exercises in Unit 1 – Exercise
2.2.4”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.2.2.pdf)
(PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.2.3 Definable Sets**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 2.3”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
2.3” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 2.3 in its entirety, and solve
Exercise 2.3.6. When you are finished, you can check your answers
against the Saylor Foundation’s [“Solutions for Exercises in Unit 1
– Exercise
2.3.6”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.2.3.pdf)
(PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.3 Elementarity**
**1.3.1 Theories and Elementarity**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Chapter 3 Introduction and Section 3.1”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Chapter 3 Introduction and Section
3.1” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read from the beginning of Chapter 3 through
the end of section 3.1, solving the two in-text exercises as you
come to them. Continue to read the proofs carefully (i.e.,
skeptically). The proof in section 3.1.11 is worthy of special
mention: it is the classic example of the so-called “back-and-forth”
argument, which is ubiquitous in model theory. Why is it called
“back-and-forth”? Can you say what the general method is? When you
are finished, you can check your answers against the Saylor
Foundation’s [“Solutions for Exercises in Unit 1 – Exercise 3.1.2
and
3.1.4”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.3.1.pdf)
(PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.3.2 Elementary Maps and Partial Isomorphisms**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 3.2”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
3.2” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 3.2 in its entirety, solving the
in-text exercises as you come to them. The importance of this
section is hard to overstate; almost everything else in model theory
depends on elementary maps, and on the ability to match parts of
structures. Notice that, in general, an elementary bijection need
not be an isomorphism. Under what conditions must it be? When you
are finished, you can check your answers against the Saylor
Foundation’s [“Solutions for Exercises in Unit 1 – Exercise 3.2.1,
3.2.3, 3.2.4, and
3.2.5”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.3.2-1.pdf)
(PDF).
This reading should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**Assignment: Università di Torino: Domenico Zambella’s A Crèchet Course in Model Theory: “Exercise 1”**Link: Università di Torino: Domenico Zambella’s*A Crèchet Course in Model Theory*: “Exercise 1” (PDF)Instructions: Click on the link to “Exercise 1” and solve the problem. When you are finished, you can check your answers against the Saylor Foundation’s “Solutions for Exercises in Unit 1 – Exercise 1” (PDF).

This assignment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**1.3.3 The Downward Löwenheim-Skolem Theorem**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 3.3”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
3.3” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read section 3.3 in its entirety, solving the
in-text exercise as you come to it. Notice how counterintuitive the
Löwenheim-Skolem Theorem is: All properties of the real numbers that
can be expressed with first-order formulas can simultaneously be
satisfied in a countable structure. In algebra, this is not so
surprising, but this is extremely restrictive for expressing
analysis. Any talk of limits is, in particular, problematic. Many
interesting parts of calculus can be reconstructed on algebraic
grounds, but the usual approach to calculus and analysis is
completely missed by the first-order theory. When you are finished,
you can check your answers against the Saylor
Foundation’s [“Solutions for Exercises in Unit 1 – Exercise
3.3.4”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.3.3.pdf) (PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.4 Compactness**
**1.4.1 Consistency**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Chapter 4 Introduction and Section 4.1”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Chapter 4 Introduction and Section
4.1” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read from the beginning of Chapter 4 through
the end of section 4.1. Don’t be put off by the “How not to read
this chapter” paragraph. Zambella’s advice on skipping the Henkin
proof is reasonable enough for the pure model theorist, but for the
present course, where we intend eventually to understand
computation, this proof is important: it is a proof by induction,
but one in which we must order the stages carefully in order to
finish in a finite time. Such constructions are at the heart of
computability. As usual, try to think of examples and near-misses
for the definitions.
This reading should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.4.2 Compactness**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 4.2”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
4.2” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read all of section 4.2, solving the in-text
exercises as you come to them. Compactness is, in an important
sense, the combinatorial heart of model theory, and the interaction
between partial isomorphisms and compactness accounts for much of
the discipline. If you have heard mention of compactness in
mathematics before, perhaps in analysis or topology, the notion is
exactly the same (and, in fact, the Compactness Theorem in model
theory can be taken as the statement that a certain topological
space is compact). The proofs are technical but important. When
you are finished, you can check your answers against the Saylor
Foundation’s [“Solutions for Exercises in Unit 1 – Exercise 4.2.8
and
4.2.9”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.4.2.pdf)
(PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**1.4.3 Monster Models**
- **Reading: Università di Torino: Domenico Zambella’s A Crèchet
Course in Model Theory: “Section 4.3”**
Link: Università di Torino: Domenico Zambella’s *A Crèchet Course in
Model Theory*: “Section
4.3” (PDF
or PS)

```
Instructions: Click on the link to the PDF or PS version of any
version of the text. Read all of section 4.3, solving the in-text
exercise as you come to it. You could think of the complex numbers
as being the example that motivates monster models. In the complex
numbers, we have solutions to every polynomial equation we could
ever think of, and so algebra can be done comfortably there – even
if it is a bit bigger than the world we find “natural.” A *monster
model* is exactly that: an awkwardly big world in which we can
comfortably find solutions to formulas. When you are finished, you
can check your answers against the Saylor Foundation’s [“Solutions
for Exercises in Unit 1 – Exercise
4.3.10”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/MA201-Solutions-1.4.3.pdf)
(PDF).
This reading should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```