A Course in Model theory^{1} Errata

K. Tent and M. Ziegler

November 28, 2014

1Cambridge University Press, 2012

We thank M. Junker, A. Berarducci, A. Blumensath, D. Müller and D. Pierce for bugreports.

• In some places, the use of the word either is not justified. Even though it should not cause confusion, remove the word in Exercise 8.2.6,

• Definition 1.2.10, Footnote 5: we replace each bound occurrence by an unused variable.

• Exercise 2.3.1, part 2: The structure is (ℚ, P_{r}, Q_{r})_{r∈ℝ}.

• In the proof of Theorem 3.1.8, b) ⇒ a), one should add that 𝔄 and 𝔅^{0} are models of T_{1} and T_{2}, respectively,

• The Morleyisation of T as discussed after Definition 3.2.1 should use 0-ary relation symbols. Otherwise it does not have quantifier elimination if T is consistent, incomplete and L has no constants.

• Exercise 3.2.1, part 1: For any T -ec-structure 𝔐…

• Theorem 3.2.5:

In c) we may assume that 𝔄 is finitely generated. This is used in later applications.

• Proof of Theorem 3.3.22, third paragraph: Replace g(a, da, …, d^{n−1}a) ≠ 0 by g(b, db, …, d^{n−1}b) ≠ 0.

• In Exercise 3.3.1, replace

⋀_{i≠j}  by  ⋀_{i,j} .

• The first pragraph of Section 4.2 misleadingly claims a connection between the Compactness Theorem and the compactness of the space of types.

• In Definition 5.2.5, do not assume that T is countable.

• Proof of Lemma 6.3.2:

Choose \overline{b}^{2i} algebraically independent over a\overline{b}^{0} … \overline{b}^{2i−1} (not only over \overline{b}^{0} … \overline{b}^{2i−1}).

• Proof of Lemma 6.3.6:

Replace

This implies that diff(a_{1}, a_{2}) only depends on tp(a_{1}, a_{2}).

by Independence of the model implies that diff(a_{1}, a_{2}) only depends on tp(a_{1}, a_{2}).

• Definition 8.1.1, part 1: … there is some m ∈ M with φ(x, m) ∈ p.

• The definition before Corollary 7.2.7 should read: We call q a forking extension if q forks over A.

• Proof of Corollary 8.1.8: the formula should read

⊧ ∃x, y φ(x, y, c) ∧ φ_{1}(x, a′, h′_{1}) ∧ φ_{2}(y, b′, h′_{2}).

• Proof of Lemma 8.3.5, last paragraph: a is a realization of p ↾ Mb.

• Exercise 8.3.5: A type p being stable should be defined as: for no formula φ(x, y) there are elements a_{0}, a_{1}, … satisfying p(x) and b_{0}, b_{1}, … such that ⊧ φ(a_{i}, b_{j}) ⇔ i < j.

• Proof of Lemma 8.4.10: … c_{1} … c_{m} ∈ acl(e).

• Proof of Theorem 8.5.10: in the second to last paragraph, it should read Let μ be the cardinal given by Weak Boundedness applied to p.

• Exercise 8.5.2: Let p ∈ s(B) and q ∈ S(B) be two different types which do not fork over A ⊆ B…

• Before Lemma 10.4.1: … induced by M and N.

• Before Theorem 10.4.8: … we say that M ∈ K_{μ} is K_{μ}-saturated if for all finite A ≤ M and finite strong extensions C of A…

• Definition B.4.10:

The reader should keep in mind that we define procyclic fields to be perfect.

• After Remark C.1.1: A pregeometry in which points and the empty set are closed, i.e., in which

cl(∅) = ∅ and cl(x) = {x} for all x ∈ X,

is called geometry.

• Discussion after Definition C.1.9: Add the sentence Note that the proof shows that tr. deg(K) ≥ 3 actually suffices.

• Append the following sentence to the proof of Lemma C.1.11:

Since c cannot be in the closure of {a_{1}, …, a_{n−2}} ∪ (A ∩ B), we know by induction that {a_{1}, …, a_{n−2}, c} is independent over B, which contradicts c ∈ B′.

• Replace the last paragraph before Exercise C.1.1 by The arguments on page 205 show also that a field of transcendence degree at least 4 is not locally modular. Just replace F by a subfield of transcendence degree 1.

• Solution to Exercise 6.2.8: … and φ(x, a_{1}) ∨ ⋯ ∨ φ(x, a_{n})…

• Solution to Exercise 7.1.3: … implies a conjunction ⋁_{ℓ<d} φ_{ℓ}(x, b) of formulas which divide over A.