Master conditions and proper posets

Let $P$ be a nonempty preorder, let $\theta$ be a regular cardinal with $P\in H_{\theta }$, and let $M$ be a countable elementary submodel of a structure $(H_{\theta },\in ,{<}_{\theta },P,\dots )$ containing all displayed parameters, where ${<}_{\theta }$ is a fixed well-order of $H_{\theta }$. A condition $q\in P$ is $(M,P)$-generic if, for every dense $D\subseteq P$ with $D\in M$, the set $D\cap M$ is predense below $q$. Spelled out in the stronger-is-smaller convention, this means

$\mathrm{\forall }r\le q\mathrm{\exists }s\in D\cap M\text{ such that }r\text{ and }s\text{ are compatible in }P\mathrm{.}$

For $p\in P\cap M$, an $(M,P)$-master condition below $p$ is an $(M,P)$-generic $q$ satisfying $q\le p$. Neither genericity nor mastery requires $q\in M$, and genericity does not require $q$ itself to belong to every dense set.

The preorder $P$ is proper in the master-condition formulation if, for every sufficiently large regular $\theta$, every such countable elementary $M$, and every $p\in P\cap M$, there is an $(M,P)$-master condition below $p$. Here “sufficiently large” means that some regular ${\theta }_0$ works for every regular $\theta \ge {\theta }_0$. Adding the well-order makes the Skolem-closure convention explicit. The equivalent club-of-models and generic-extension formulations are assertions, not definitions, and are proved in the next item.

This item only fixes predicates and quantifiers, so it makes no selection and uses no instance of Choice. Existence of the countable elementary models and of master conditions is invoked only by later theorems under their declared axiom bases.