Forcing formulas in Fraïssé structures and classes

A. T. Nurtazin

Algebra i logika, Tome 57 (2018) no. 5, pp. 567-586

Résumé

We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class $\mathscr F$. Both known and some new necessary and sufficient conditions are derived under which a given structure $\mathscr M$ will be a forcing structure. A formula $\varphi$ is forced at $\bar a$ in an infinite structure $\mathscr M\Vdash\varphi(\bar a)$ if it is forced in $\mathscr F(\mathscr M)$ by some finite substructure of $\mathscr M$. It is proved that every $\exists\forall\exists$-sentence true in a forcing structure is also true in any existentially closed companion of the structure. The new concept of a forcing type plays an important role in studying forcing models. It is proved that an arbitrary structure will be a forcing structure iff all existential types realized in the structure are forcing types. It turns out that an existentially closed structure which is simple over a tuple realizing a forcing type will itself be a forcing structure. Moreover, every forcing type is realized in an existentially closed structure that is a model of a complete theory of its forcing companion.

Détail

Keywords: forcing method, forcing structure, forcing type, existentially closed structure, existentially closed companion.
Mots-clés : Fraïssé class