4 1. COMBINATORIAL GEOMETRIES AND INFINITARY LOGICS

Exercise 1.1.3. If 𝐺 is a homogeneous geometry, 𝑋, 𝑌 are maximally inde-

pendent subsets of 𝐺, there is an automorphism of 𝐺 taking 𝑋 to 𝑌 .

The most natural examples of homogeneous geometries are vector spaces and

algebraically closed fields with their usual notions of closure. The crucial properties

of these examples are summarised in the following definition.

Definition 1.1.4. (1) The structure 𝑀 is strongly minimal if every first

order definable subset of any elementary extension 𝑀′ of 𝑀 is finite or

cofinite.

(2) The theory 𝑇 is strongly minimal if it is the theory of a strongly minimal

structure.

(3) 𝑎 ∈ acl(𝑋) if there is a first order formula with parameters from 𝑋 and

with finitely many solutions that is satisfied by 𝑎.

Definition 1.1.5. Let 𝑋,𝑌 be subsets of a structure 𝑀. An elementary iso-

morphism from 𝑋 to 𝑌 is 1-1 map from 𝑋 onto 𝑌 such that for every first order

formula 𝜙(v), 𝑀 ∣ = 𝜙(x) if and only if 𝑀 ∣ = 𝜙(𝑓x).

Note that if 𝑀 is the structure (𝜔,𝑆) of the natural numbers and the successor

function, then (𝑀,𝑆) is isomorphic to (𝜔 − {0},𝑆). But this isomorphism is not

elementary.

The next exercise illustrates a crucial point. The argument depends heavily on

the exact notion of algebraic closure; the property is not shared by all combinato-

rial geometries. In the quasiminimal case, discussed in the next chapter, excellence

can be seen as the missing ingredient to prove this extension property. The added

generality of Shelah’s notion of excellence is to expand the context beyond a com-

binatorial geometry to a more general dependence relation.

Exercise 1.1.6. Let 𝑋,𝑌 be subsets of a structure 𝑀. If 𝑓 takes 𝑋 to 𝑌 is

an elementary isomorphism, 𝑓 extends to an elementary isomorphism from acl(𝑋)

to acl(𝑌 ). (Hint: each element of acl(𝑋) has a minimal description.)

The content of Exercise 1.1.6 is given without proof on its first appearance

[BL71]; a full proof is given in [Mar02]. The next exercise recalls the use of

combinatorial geometries to study basic examples of categoricity in the first order

context.

Exercise 1.1.7. Show a complete theory 𝑇 is strongly minimal if and only if

it has infinite models and

(1) algebraic closure induces a pregeometry on models of 𝑇 ;

(2) any bijection between 𝑎𝑐𝑙-bases for models of 𝑇 extends to an isomorphism

of the models.

Exercise 1.1.8. A strongly minimal theory is categorical in any uncountable

cardinality.

1.2. InfinitaryLogic

Infinitary logics 𝐿𝜅,𝜆 arise by allowing infinitary Boolean operations (bounded

by 𝜅) and by allowing quantification over sequences of variables length 𝜆. Various

results concerning completeness, compactness and other properties of these logics

were established during the late 1960’s and early 1970’s. See for example [Bar68,