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MATH 144: COURSE NOTES
WILL BONEY
Contents
4. February 1
5. February 3
6. February 5
References
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6
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4. February 1
We’re going to momentarily cover some stuff not as presented in [TZ]. This is
essentially [M, Section 1.3].
4.1. Definability.
What do we do with model theory? There are many answers to this question,
some of which we’ll get to in this class. At a very basic level, we can take a
structure that we want to study R and view it as a model theoretic structure.
Then we can use logic to pick out the nice subsets of it, called definable.
Definition 4.1. Fix an L-structure M .
(1) Given φ(x, y) and m ∈ |M |, we write
φ(M, m) := {a ∈ |M | : M φ(a, m)}
We might write φ(M n , m) for this if we want to emphasize that x is of
length n.
(2) We say that X ⊂ |M |n is definable iff there is some φ(x, y) and m ∈ |M |
such that X = φ(M, m). In this case, we say φ(x, m) defines X. To specify
the parameters, we say that X is A-definable or definable over A if there
is a definition with parameters from A. The special case of ∅-definable or
0-definable means that there are no parameters.
(3) We say that a function or relation is definable iff it is definable when considered as a subset of the universe.
Date: February 2, 2016.
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WILL BONEY
Example 4.2.
(1) Let Z = {Z, 0, +, <}. Then we can define N by φ(x) =
“x = 0 ∨ x > 0”.
(2) Let R = {R, 0, 1, +, ∗, −, 1· } be the field of real numbers. The ordering on
R is definable by φ(x, y) ≡ “∃z(x + z 2 = y)00 .
(3) Let F be a field and set M = {F [X], +, −, ×, 0, 1} be the ring of polynomials
in one variable over F . Then F is definable in M as the set of units. In
particular, F = φ(M ), where φ(x)
= 1)00 .
Pm≡ “∃y(xy
i
(4) Let R be a ring and p(X) =
i=0 ai X ∈ R[X] be a polynomial over
R. The zeros of this polynomial are definable over (any set containing)
{a0 , . . . , am }. Set
φ(x, y0 , . . . , ym ) = “ym xm + ym−1 xm−1 + · · · + y0 = 0”
where xm is an abbreviation for x × · · · × x m times. Then the zeros are
exactly φ(R, a).
(5) Let Q = {Q, +, −, ×, 0, 1}. Set
• φ(x, y, z) ≡ “∃a, b, c(xyz 2 + 2 = a2 + xy 2 − yc2 )”
• ψ(x) ≡ “∀y, z ([φ(y, z, 0) ∧ (∀w(φ(y, z, w) → φ(y, z, w + 1)))] → φ(y, z, x)) ”
Julia Robinson (see [FlWa91]) showed that ψ(x) defines the integers in Q.
One theme of definability is that proving wether or not a set is definable actually
requires some knowledge about the structure you’re working in. See the homework
and imagine that I had said nothing about Lagrange’s Theorem.
Definable sets are better behaved than sets in general, but still have some very
nice closure properties. Here’s a purely semantic way about thinking of definable
sets.
Proposition 4.3 ( [M].1.3.4). Let M be an L-structure. Suppose that Dn is a
subset of |M |n and {Dn | n ∈ N} is the smallest collection of such subsets such
that
• M n ∈ Dn ;
• for each n-ary f ∈ L, the graph of f M := {(m, m) | f M (m) = m} ∈ Dn+1 ;
• for each n-ary R ∈ L, RM = {m | M R(m)} ∈ Dn ;
• for each i, j ≤ n, {(x1 , . . . , xn ) ∈ |M |n : xi = xj } ∈ Dn ;
• if X ∈ Dn , then M × X ∈ Dn+1 ;
• each Dn is closed under complement, union, and intersection;
• if X ∈ Dn+1 and π is the projection map (m1 , . . . , mn+1 ) 7→ (m1 , . . . , mn ),
then π“X ∈ Dn ; and
• if X ∈ Dn+m and b ∈ |M |m , then {a ∈ |M |n : (a, b) ∈ X} ∈ Dn ,
then {Dn | n ∈ N} are exactly the definable sets of M .
The proof of this is straightforward and I’m not too interested in giving it. Essentially, the first two clauses correspond to the basic relations; complements, etc. correspond to negations, etc.; and projection corresponds to existential quantification.
MATH 144: COURSE NOTES
3
Then a little more work must be done to account for composition/substitution and
reordering of variables.
Unfortunately (or maybe fortunately), not all subsets are definable.
Proposition 4.4. Every infinite structure M in a countable language has undefinable subsets.
Actually, nothing about this requires countability, just that kM k ≥ |LM |.
Proof: There are ℵ0 many formulas and kM k<ω = kM k many tuples. Thus
there are ℵ0 × kM k = kM k many possible definitions for sets, so this is an upper
bound for the number of definable sets (turns out we have exactly this many).
On the other hand, there are 2kM k > kM k many subsets of M .
†
Okay, but that’s not constructive. In particular, it doesn’t help us determine if
a particular subset is definable. The following is useful for this.
Proposition 4.5 ( [M].1.3.5). Let M be an L-structure. If X ⊂ |M |n is Adefinable, then every automorphism of M that fixes A pointwise also fixes X setwise.
Proof: Let f ∈ AutA M := {f : M ∼
= M | ∀a ∈ A, f (a) = a. We want to show
00
that such that X = f X := {f (x) | x ∈ X}. By A-definability, there is a ∈ A
and φ(x, y) such that X = φ(M, a). Let m ∈ X; then M φ(m, a). Hitting this
with f gives M φ(f (m), a), so f (m) ∈ X. Note that f − 1 ∈ AutA M as well, so
f −1 (m) ∈ X. Thus, X = f 00 X.
†
We can use this to show that the reals are not definable in the complexes (as a
field).
Proposition 4.6 ( [M].1.3.6). R is not definable in {C, +, ×, 0, 1}.
It’s necessary to indicate the language because it obviously is definable if we
add a unary predicate that consists of the real numbers.
Proof: Suppose that R were definable. Then it would be definable over some
finite A ⊂ C. We can find r, s ∈ C that are algebraically independent1 over A
with the extra property that r ∈ R and s 6∈ R. From Galois theory, we know that
there is f ∈ AutA C that sends r to s. So we have found an automorphism of C
that fixes A pointwise, but doesn’t fix R. This contradicts Proposition 4.5.
†
A converse to Proposition 4.5 would be nice. However, that’s impossible in
general. If M is rigid (no automorphisms), then that would imply that every set is
definable which we know is not the case. When do we have automorphisms? Well,
C has lots of automorphisms. This turns out to be an example of a model theoretic
1This
A.
means that they are not roots of any polynomial with coefficients in the field generated by
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WILL BONEY
property of models called “saturated.” When we construct saturated models, they
will have lots of automorphisms and that will give us a partial converse.
Proposition 4.7 ( [M].4.3.25). Let M be saturated and A ⊂ |M | with |A| < kM k.
Suppose X ⊂ |M |n is M -definable. Then X is A-definable iff every automorphism
of M that fixes A pointwise also fixes X setwise.
4.2. Interpretation.
The following is a motivating example.
Example 4.8. C is bigger than R. However, if we have R (as a field), we can
reason about C (as a field) in the way you are all familiar with. Set
X = {(a, b) | a, b ∈ R}
φ+ (a, b; c, d; e, f ) ≡ “a + c = e ∧ b + d = f ”
φ× (a, b; c, d; e, f ) ≡ “ac − bd = e ∧ bc + ad = f ”
Both are definable function from X×X to X and it is easy to see that (X, φ+ , φ× ) ∼
=
(C, +, ×).
This motivates the notion of an interpretation.
Definition 4.9. Let M be a τ -structure and N be a τ 0 structure. M is definably interperable in N iff we can find a definable X ⊂ |N |n , a definable function
φF (xF , aF ) on X with aF ∈ N of the same arity2 as F for each function F ∈ τ ,
and definable relations φR (yR , aR ) on X with aR ∈ N of the same arity as R for
each relation R ∈ τ such that
M∼
= {X, φF (xF , aF ), φR (yR , aR ) | F, R ∈ τ }
Let’s look at a more interesting example. We will outline how to interpret any
order in a graph; the full details are in [M, Section 1.3].
We want to describe a map from orders I to graphs GI that allows one to recover
the order from the graph in a uniform way. Define the vertices of GI as follows:
• For each a ∈ I, add vertices a, xa1 , xa2 , xa3 .
• For each a <I b, add vertices y1a,b , y2a,b , y3a,b
Define the edges of GI as follows:
• For each a ∈ I,
– xa1 , xa2 , xa3 are all connected
– a and xa1 are connected
• For each a <I b,
– y1a,b is connected to a, y2a,b , y3a,b
– y2a,b is also connected to b.
2The
length of x here is the arity of F multiplied by n.
MATH 144: COURSE NOTES
5
That G is a graph of this form is first-order axiomatizable.
Given a graph of this form, we can read I off in the following way: the elements
of I are precisely the vertices that are connected to a triangle: φI (G), where
!
^
φ(x) ≡ “∃x1 , x2 , x3 xEx1 ∧
xi Exj ”
1≤i6=j≤3
We can define the relation < on these vertices by saying that a < b iff there is a
path of length 2 from a to b and the first vertex in the path has another vertex
coming off of it: φ< (G2 ), where
φ(x, y) ≡ “∃y1 , y2 , y3 (xEy1 Ey2 Ey ∧ y1 Ey3 ) ”
Beyond definable interpretations, there is the notion of interprability. This
expands the notion of definable interprability by specifying a definable equivalence
relation on the definable set, and using that as the universe for the interpretation.
The following is a formal definition, although we won’t use it or give it in class.
Definition 4.10. Let M be a τ -structure and N be a τ 0 structure.
(1) Given an equivalence relation E, a function f is E-invariant iff, whenever
x1 Ey1 , . . . , xn Eyn , we have f (x1 , . . . , xn )Ef (y1 , . . . , yn ).
(2) M is interperable in N iff we can find a definable X ⊂ |N |n , a definable
equivalence relation E, a definable E-invariant function φF (xF , aF ) on X
with aF ∈ N of the same arity as F for each function F ∈ τ , and definable
E-invariant relations φR (yR , aR ) on X with aR ∈ N of the same arity as
R for each relation R ∈ τ such that
M∼
= {X/E, φ∗ (xF , aF ), φ∗ (yR , aR ) | F, R ∈ τ }
F
R
What’s the difference between definably interperable and interperable? What’s
the deal with that equivalence relation? To see, let’s look at an example.
Example 4.11. Let G Tgrp (i. e., is a group) and let H ⊂ G be a definable,
normal subgroup (say by φ(x, g)). I would like to talk about G/H. The best I can
do is that G/H is interperable in G: set X = G, xEy iff ∃h (φ(h, g) ∧ x = yh),
and φ· (x, y, zg) is ∃h (φ(h, g) ∧ xy = zh). Then φ· is an E-invariant function and
E is an equivalence relation. Moreover
G/H ∼
= (X/E, φ∗ )
+
However, this is less than desirable. The model theory of groups is big and can
say lots of things about definable subgroups. Here are some examples:
Theorem 4.12 ( [M].7.1.2, Macintyre). Every ω-stable group satisfies the Descending Chain Condition for definable subgroups.
Theorem 4.13 ( [M].7.2.11, Reinke). If G is an infinite ω-stable group with no
proper definable infinite subgroups, then G is abelian.
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WILL BONEY
However, this study is intimately connected with the notion of definability.
Thus, the fact that we can have a definable, normal subgroup and have the quotient group be undefinable is problematic. There is a way to deal with this that
we might cover later (as a teaser, it is called elimination of imaginaries because it
generalizes the idea of adding imaginary numbers to the reals).
Here is an example of how this plays out in current research.
Definition 4.14.
(1) A model M is minimal iff every definable subset is finite
or cofinite.
(2) A model M with < in the language is o-minimal iff every definable subset
of3 M is a finite union of points and intervals.
Minimality means that the definable subsets are precisely those that can be
defined with just equality, and o-minimality means that the definable subsets are
precisely those that can be defined with equality and the order and equality. These
sound like really boring theories, but there are some nice theories that fall into
these categories:
• algebraically closed fields and vector spaces are minimal
• several expansions of (R, 0, 1, +, ×, <) are o-minimal, including adding ex
or all bounded analytic functions
o-minimality in particular is a very active area of research. As detailed in a
book by Lou van den Dries [?] of the same name, o-minimality is a way to realize
a framework suggested by Grothendieck of developing a tame topology for the
real numbers. With the order, you can define the standard topology on the real
numbers and definable subsets give you a control over describing the subsets that
you can describe.
Here’s a nice result about o-minimal structures:
Theorem 4.15. Every definable function in an o-minimal structure is piecewise
monotonic (continuous, differentiable, etc).
5. February 3
The ordering of the next bit is a little tricky. The next three core concepts
we introduce–Compactness, Elementary Substructure, and Complete Theories–all
have some dependencies between the key results that we would want to prove
about them. This makes it a little tricky to pick the order of them. So things
might be a little strange, or back track a bit, but it’ll all be clear in the end. The
pay off will be an easy proof of Ax-Grothendieck, if that’s the sort of thing that
you’re into.
3That
is, in one variable.
MATH 144: COURSE NOTES
7
5.1. Compactness.
We talked about this already.
Definition 5.1. A theory T is finitely satisfiable iff every finite T0 ⊂ T is satisfiable.
Theorem 5.2. A theory is satisfiable iff it is finitely satisfiable.
Gödel proved this in 1930 for countable theories and Mal’tsev extended this to
all theories in 1936.
Remember:
• Consistent is the same as satisfiable.
• Our “proof” of this is that proofs are finite, so a proof of a contradiction
uses only finitely many axioms.
This is THE theorem of first order logic. It is amazing. This will be made formal
in Lindstrom’s Theorem 6.10, but for now we have applications.
First, we can finally show that certain ideas are not first order expressible. The
basic outline is this:
• Identify some property P and suppose that the theory T axiomatizes this
property.
• Find expansion T + of T that express violating P .
• Show that T + is consistent by modeling it’s approximations by structures
that satisfy P .
• Conclude that there is a model of T that does not satisfy P .
Proposition 5.3. The notion of being finite is not first-order axiomatizable. More
precisely, if T has arbitrarily large finite models, then T has infinite models.
Proof: Suppose that T has arbitrarily large finite models. Set L+ := L(T )∪{cn |
n ∈ N} and set
T + := T ∪ {cn 6= cm | n 6= m ∈ N}
Any model of T + will a) model T and b) be infinite. So we must show that T + is
consistent.
Let T0 ⊂ T + be finite. Let N be the maximum index of a cn that appears in T0 .
Then
T0 ⊂ T ∪ {cn 6= cm | n 6= m ≤ N }
Let M 0 be a model of T that has at least N many elements of it. Then expand
M 0 to an L+ -structure M + by interpreting the N -many cn ’s as distinct elements.
So M + |= T0 .
Thus T has infinite models.
†
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WILL BONEY
Contrast this with the fact that, for each n ∈ N, it is a first order property
to say that a structure has n elements in it. We can also say that a structure is
infinite with a theory, but the negation of this property is not first order.
We’ll do a few more examples in class, and then there are more on the homework.
Proposition 5.4. The notion of being a torsion group is not first-order axiomatizable.
Proof: Suppose that T is a theory extending Tgrp (so L(T ) extends Lgrp ) such
that for every n, there is Mn T and gn ∈ Mn such that gn has order at least n
(in Mn ). Then set L+ := L(T ) ∪ {c} and set
T + := T ∪ {cn 6= e | n ∈ N}
where xn is an abbreviation for the term of x group operated by itself n times.
Any model of T + will a) model T and b) have an element of infinite order. So we
must show T + is consistent.
Let T0 ⊂ T + be finite. Let N be the maximum number n so “cn 6= e00 ∈ T0 .
Then
T0 ⊂ T ∪ {cn 6= e | n ≤ N }
†
Then (MN +1 , gN +1 ) T0 .
Definition 5.5. An order (I, <) is well-ordered iff there is no infinite, descending
chain. That is, there is no hxn ∈ I | n ∈ Ni such that xn > xn+1 .
Proposition 5.6. The notion of being well-ordered is not first-order axiomatizable.
Proof: Let T be a theory extending Tord with infinite models. Set L+ :=
L(T ) ∪ {cn | n ∈ N} and set
T + := T ∪ {cn > cn+1 | n ∈ N}
Any model of T + will a) model T and b) have an infinite decreasing sequence
witnessed by the interpretations of the cn . So we must show T + is consistent.
Let T0 ⊂ T + be finite. Let N be the maximum number so cN ∈ L(T0 ). Let
M be an infinite model of T . Since M is an infinite linear order, it contains arbitrarily long decreasing finite sequences. Interpret c1 , . . . , cN in M as one of these
sequences m1 , . . . , mN of length N . Then (M, c1 , . . . , cN ) T0 . Thus T + is consistent.
†
Maybe a little surprisingly, we can also use compactness to show that a property
is first order.
Lemma 5.7. Given a finite graph G, there is a sentence φ¬G in Lgraph such that
a graph models φ¬G iff G does not appear as an induced subgraph of it.
MATH 144: COURSE NOTES
9
Proof: Enumerate the vertices of G as {vn | n ≤ N } and set V := {(i, j) |
vi and vj are connected in G}. Define φ¬G to be
_
∀x1 , . . . , xN
¬xi Exj
(i,j)∈V
†
This works.
Lemma 5.8. A graph is k-colorable iff every finite induced subgraph is k-colorable.
Proof: One direction is obvious. So suppose that G = (V, E) is an infinite
graph for which every finite induced subgraph is k-colorable. Set
L := Lgraph ∪ {Ri (·) | i < k} ∪ {cv | v ∈ V }
Remember the natural numbers start at 0. Set T to be
(1) Tgraph
W
(2) ∀x i<k Ri (x)
V
(3) ∀x, y xEy → i<k ¬(Ri (x) ∧ Ri (y))
(4) cv Ecw for every v, w ∈ V for which vEw
(5) ¬(cv Ecw ) for every v, w ∈ V for which ¬(vEw)
Then we have two claims:
Claim 1: T is consistent.
Proof: Take a finite subset T0 of T . Then any model of T0 must be a graph that
is k-colorable (as determined by the Ri ) and have a copy of the induced subgraph
H formed by the vertices {v ∈ V | cv ∈ L(T0 )}. In particular, H expanded by
predicates for the colors and constants for each vertex models T0 . By compactness,
T is consistent.
Claim 2: A model of T gives a k-coloring of G.
Proof: Let M T . Define a map f : V → |M | by v 7→ cM
v . Because T connect
constants iff the vertices they represent are connected, this is actually an Lgraph
isomorphism f : G → M Lgraph . Then we can pull back the coloring on M by
giving v color i iff M Ri (cv ). By the axioms of T , this is a k-coloring.
†
Theorem 5.9. There is a theory T≤k in Lgraph such that a graph is k-colorable iff
it models T≤k .
Proof: We put together the previous two lemmas. Set G>k be the set of
(isomorphism representatives of) all finite graphs that are not k-colorable. Set
T≤k = {φ¬G | G ∈ G>k }. Suppose that H T . We claim that every finite induced
subgraph (substructure) of H is k-colorable. If not, then this graph G would be
in G>k , so would not appear as an induced subgraph of any model of φ¬G . Thus,
by Lemma 5.8, H is k-colorable.
†
Notice that we’ve given a first order axiomatization to being k-colorable, not
having chromatic number k. See the homework.
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WILL BONEY
We can also use the compactness theorem to prove the finite version of Ramsey’s
Theorem from the infinite one. See the homework.
6. February 5
The technique of adding the constants to the language from Lemma 5.8 is very
useful.
Definition 6.1. Suppose M T and A ⊂ |M |. Then L(A) := L(T ) ∪ {ca | a ∈ A}
and set the atomic diagram of A to be
ADM (A) := {φ(ca ) | a ∈ A and φ is an atomic formula such that M φ(a)}
We write L(M ) instead of L(|M |).
The following are straightforward:
Proposition 6.2. Let M T and A ⊂ |M |.
(1) ADM (A) {φ(ca ) | a ∈ A and φ is an atomic formula such that M φ(a)}
(2) If (N, na )a∈A ADM (A), then the map a 7→ na preserves quantifier-free
formulas.
(3) In particular, if (N, nm )m∈|M | ADM (M ), then the map m 7→ nm is an
L(T )-embedding.
Proof: You’ll prove a stronger version of this in your homework.
†
One of the central notions in model theory is that of a type. Roughly, a type is
description of an element using some parameters.
Definition 6.3. Let M be an L-structure and A ⊂ |M |. Let x be a (finite)
sequence of free variables.
• A type over A p(x) is a collection of L(A) formulas that is consistent with
ThL(A) (M, a)a∈A .
• Given b ∈ |M |, the type of b over A in M is
tp(b/A; M ) := {φ(x, a) | a ∈ A and M φ(b, a)}
• Given a type p(x) over A, we say that some c ∈ |M | realizes p iff, for every
φ(x, a) ∈ p(x), M φ(c, a). This is written c p. We say that M realizes
p iff it contains a tuple that does so; we write this M p.
There’s some worry about formalism that we don’t fuss about too much, but
should mention. There are two ways of imagining types:
• A type p(x) over A is a theory in the language L ∪ {ca | a ∈ A} ∪ cx .
M
A realization of a type is some structure (M, cM
a , cx )a∈A that models this
theory, where the actual realization of the type is the interpretation of the
constant cx .
MATH 144: COURSE NOTES
11
• A type p(x) over A is a collection of L-formulas with free variable x and
parameters from A. A realization of a type is some structure M that
contains a nice copy of A and the realization of the type is m ∈ |M | that
satisfies all of the formulas.
People tend to treat it as the first, mainly because there’s a lot loss formal baggage
floating around.
Example 6.4.
(1) Given A ⊂ |M | and a ∈ A, the type of a over A is determined by the inclusion of the formula “x = a00 . Types like this are called
algebraic (actually a slightly larger class) and are boring.
(2) Suppose (F, 0, 1, +, ×) is a field and A ⊂ F . What are the types over A?
This boils down to what can one say about possible field elements using A?
By the above, one can describe the elements of A and even the elements
of the field F (A) generated by A. Beyond that, one could also describe
something as the root of a polynomial whose coefficients are in F (A). One
could even describe something transcendental over A by saying that it is
not the root of any polynomial
ptrans (x) = {an xn + . . . a0 6= 0 | n ∈ N and ai ∈ A}
It turns out that these are precisely the types over A.
(3) Let TDLO be the Lord -theory of dense linear orders. Let A ⊂ M TDLO .
The types over A correspond to the Dedekind cuts of (A, <).
We’ve defined type in two different ways, but that’s alright.
Proposition 6.5. Given b ∈ |M | and A ⊂ |M |, tp(b/A; M ) is a type over A.
Proof: We just have to argue that tp(b/A; M ) is consistent with the given
A
= a–
theory. Expand M to an L(A)-structure MA in the intended way–i. e., cM
a
and expand L(A) further by adding a constant ci for each bi . Then (M, a, b)a∈A
models
p(cb ) ∪ ThL(A) (M, a)a∈A
as desired.
†
The whole point of this is that, if we have a type, then we can find a realization
of it.
Proposition 6.6. Suppose M is an L-structure. If p(x) is a type over M , then
there is N ⊃ M that realizes p.
Proof: By definition, p(c) ∪ ThL(M ) (M, m)m∈M is consistent. So let N + be a
model of it. By Proposition 6.2.(3), this gives an embedding f : M → N + L
+
by m maps to cN
m . From the second part of Homework 1.(9), we can find an
L-structure M ∗ ⊃ M and L-isomorphism f ∗ : M ∗ ∼
= N + L. Let n ∈ |N + | be
12
WILL BONEY
the interpretation of the constants c. We claim that f −1 (n) realizes p. For each
φ(x, m) ∈ p,
+
N + φ n, cN
m
N + L φ (n, f (m))
M ∗ φ f −1 (n), m
So f −1 (n) satisfies each of the formulas in p.
†
This has a very nice corollary. We will see several variations on this theme that
culminates in the Löwenheim-Skolem Theorem ??.
Corollary 6.7. Every structure has a proper extension.
Proof: Define p(x) = {x 6= m | m ∈ |M |}. Then p defines the type of an element not in M . By Proposition 6.6, there is N sup M that realizes p. Let n ∈ |N |
be such a realization. Then n 6∈ |M |, so N is a proper extension.
†
Corollary 6.8. Let T be a theory and M T . Then, for every cardinal κ, there
is some N T of size at least κ such that N ⊃ M .
Since we have covered a primer on set theory yet, take the size bit to mean
that, for every set X, we can find a model N as described with an injection
f : X → |N |. In the proof below, replace κ with X and the map x ∈ X 7→ cN
x will
be the injection.
Proof: Define L+ to be L(T ) ∪ {cm : m ∈ |M |} ∪ {ci : i < κ} (and make them
distinct) and define
T + = T ∪ ADM (M ) ∪ {ci 6= cj : i 6= j < κ}
This is consistent: given any finite subset of it, we can expand (M, m)m∈|M | to
include whatever finitely many ci ’s are in the language by interpreting them as
distinct elements. Any model of T + works as N by Proposition 6.2.(3) after renaming.
†
Here’s another application of types.
Example 6.9. Let R = {R, +, ×, 0, 1, <}. Define a type over the empty set
p(x) = {x > 1 + . . . (n − times) · · · + 1 | n ∈ N}
Let R∗ extend R, model Th(R), and realize p(x). Then we have constructed an
extension of the real numbers that has an infinite element!! That is, there is
N ∈ R∗ such that
R∗ N > 1 + · · · + 1
MATH 144: COURSE NOTES
13
for every n ∈ N. If we look at the multiplicative inverse (which exists because R
and R∗ are fields), then this is an infinitesimal element!!! We will do more with
this later.
A final note: I mentioned that compactness is the main theorem of first order
logic, and the following amazing theorem, due to Per Lindström, says that I am
half right in a very precise way.
Theorem 6.10. If L is a logic such that
(1) L is at least as expressible as first-order logic;
(2) every L-sentence in a countable language has a countable model4; and
(3) L satisfies the Compactness Theorem,
then L is first-order logic.
Put another way, first-order logic is the strongest that satisfies the Compactness
Theorem and the downward part of the Löwenheim-Skolem Theorem.
This theorem makes reference to an abstract notion of a logic, which we won’t
define formally. Informally, it turns out to be precisely what you expect after
working with different logics: a logic consists of a way of associating to each
language L a set of sentences and a satisfaction relation between L-structures and
L-sentences in a coherent way. Note there is no mention of formulas, but formulas
can be seen as sentences in a language that adds constants for the variables.
Next week we introduce the notion of elementary substructure ≺ and argue that
(ModT, ≺) is the “right” category to do model theory in. For now, you’ve got some
homework that explores what happens in the category (ModT, ⊂).
References
[TZ]
Katrin Tent and Martin Ziegler, A Course in Model Theory.
[M]
David Marker, Model Theory: An introduction.
[BaLa71] John Baldwin and Alistair Lachlan, On strongly minimal sets, The Journal of Symbolic
Logic 36 (1971), 79-96.
[ErRa56] Pál Erdős and Richard Rado, A partition calculus in set theory, Bulletin of the American Mathematical Society 62 (1956), 427-489.
[FlWa91] D. Flath and S. Wagon, How to pick out the integers in the rationals: An application
of number theory to logic, American Mathematical Monthly 98 (1991), 812-823.
[Mor65] Michael Morley, Categoricity in power, Transactions of the American Mathematical
Society 114 (1965), 514-538.
[Rob61] Abraham Robinson, Non-standard Analysis.
[Sh74] Saharon Shelah, Categoricity of uncountable theories, Proceedings of the Tarski Symposium, 1971 (1974), 187-203.
E-mail address: [email protected]
4This
is called the Löwenheim-Skolem property after the Löwenheim-Skolem Theorem ??