In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, a branch of mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, the Hrushovski construction generalizes the Fraïssé limit

Age (model theory)

In model theory, the age of a structure A is the class of all finitely generated structures which are embeddable in A . This concept is central in the construction of a Fraïssé limit....

 by working with a notion of strong substructure rather than . It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic. The specifics of determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski

Ehud Hrushovski

Ehud Hrushovski is a mathematical logician. He is a Professor of Mathematics at the Hebrew University of Jerusalem.His father, Benjamin Harshav, is Emeritus Professor in Yale University and Tel Aviv University to Comparative Literature and a poet....

to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.

Three conjectures

The initial applications of the Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have:

• Lachlan's Conjecture Any stable -categorical theory is totally transcendental.

• Zil'ber's Conjecture Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.

• Cherlin's Question Is there a maximal (with respect to expansions) strongly minimal set?

The construction

Let L be a finite relational language. Fix C a class of finite L-structures which are closed under isomorphisms and
substructures. We want to strengthen the notion of substructure; let
be a relation on pairs from C satisfying:

• implies .
• and implies
• for all .
• implies for all .
• If is an isomorphism and , then extends to an isomorphism for some superset of with .

An embedding is strong if .

We also want the pair (C, ) to satisfy the amalgamation property: if then there is a
so that each embeds strongly into with the same image for
.

For infinite , and , we say iff for
, . For any , the
closure of (in ), is the smallest superset of
satisfying .

Definition A countable structure is a (C, )-generic if:

• For , .
• For , if then there is a strong embedding of into over
• has finite closures: for every , is finite.

Theorem If (C, ) has the amalgamation property, then there is a unique (C, )-generic.

The existence proof proceeds in imitation of the existence proof for
Fraïssé limits. The uniqueness proof comes from an easy back and forth
argument.