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A filter is a nonempty upward-closed subset $G$ of a partial order such that every two members have a common lower bound in $G$. With standard notation for forcing, lower means stronger, so these are exactly the directed filters used to define a generic filter. The projection of a product-generic filter uses these properties before genericity of its factor filters is proved.

Use standard notation for forcing: $q\le p$ means that $q$ is stronger than $p$. A closed forcing order is $\lambda$-closed when every decreasing sequence $⟨p_{\xi }:\xi <\delta ⟩$ with $\delta <\lambda$ has a common lower bound:The chain condition for forcing isAn antichain in a forcing order consists of pairwise incompatible forcing conditions; incompatibility means having no common stronger extension. In particular, the ${\aleph }_1$-chain condition is the countable chain condition for forcing. These properties and their sequences or antichains are computed inside the ground model $M$.

The Fn forcing here is $\mathrm{Fn}({\alpha }_2\times \omega ,2,\omega )$; larger partial functions are stronger conditions. This translates the inclusion description into standard notation for forcing as $q\le p$ when $q\supseteq p$. Compatible members of the generic filter agree wherever both are defined, sois a function. For every $(\xi ,n)\in {\alpha }_2\times \omega$, the conditions whose domains contain that coordinate form a dense subset of a forcing order. This requirement belongs to $M$, and the generic filter meets it. Consequently $F:{\alpha }_2\times \omega \to 2$ is total.

Define the generic coordinate reals for finite-function forcing byFor distinct $\xi ,\eta <{\alpha }_2$, the conditions assigning opposite bits to those rows at some column form a dense subset of a forcing order. Given any finite condition, choose a column absent from both rows and extend by assigning zero to one coordinate and one to the other. Genericity supplies such a condition in $G$, so $a_{\xi }\ne a_{\eta }$.

The function $\xi \mapsto a_{\xi }$ is formed inside $M[G]$ by separation and replacement from $F$. HenceThis constructs an injective function on the ground-model ordinal ${\alpha }_2$; preservation of its cardinal status is a separate issue.

Use standard notation for forcing: $q\le p$ means that $q$ is stronger. A generic filter is nonempty, upward closed and downward directed, and meets every dense subset of a forcing order in the ground model. Names below are forcing names in $M$, with pairs ordered as $(\text{name},\text{condition})$.

The semantic forcing relation isHere ${\tau }^K=\mathrm{val}(\tau ,K)$ is the evaluation of a forcing name. Countability of $M$ supplies such generic filters through every condition by the Rasiowa–Sikorski lemma.

Define the syntactic forcing relation by mutual well-founded recursion on the forcing names for the atomic cases, then induction on the first-order formula. Its membership clause isFor equality, first abbreviateand set $p{⊩}^{\ast }\sigma =\tau$ exactly when both inclusions hold. Each recursive call lowers the rank of at least one forcing name without raising the other. This makes the mutual recursion well-founded.

For a basis of connectives consisting of logical conjunction, negation and existential quantification, the remaining clauses areThe last line is the existential clause of syntactic forcing; witnesses need only occur densely, rather than be forced by $p$ itself with one preselected name. Other connectives are defined by logical abbreviations. The recursion gives a definable relation inside $M$ for each fixed first-order formula, with quantification over the class of its names. If $\mathbb{P}$ has no greatest element, use canonical forcing names $\check{a}=\{(\check{b},p):b\in a,p\in \mathbb{P}\}$, which still evaluate to $a$. It also proves monotonicity: strengthening a condition preserves what it forces. The forcing theorem identifies the two relations and supplies the truth lemma.

Use standard notation for forcing, so $q\le p$ means that $q$ is stronger. Let $\check{p}$ be the canonical forcing name for the ground-model condition $p$. DefineHere $q\perp p$ means that they are incompatible forcing conditions. The checks and their collection are formed by recursion and Axiom schema of replacement in $M$, and the displayed set is selected by axiom schema of separation, so this is a forcing name in $M$.

By evaluation of a forcing name,If $p\in G$, directedness of the generic filter gives a common stronger condition for $p$ and each $q\in G$, so $p$ is absent from this value.

Conversely, for fixed $p\in \mathbb{P}$ the setis a dense subset of a forcing order belonging to $M$. A condition incompatible with $p$ is already in it, and one compatible with $p$ has a common extension below $p$. Genericity supplies $q\in G\cap D_p$. If $p\notin G$, upward closure of $G$ rules out $q\le p$, hence $q\perp p$. ThereforeThis forcing name for the complement of a generic filter works without a separativity assumption on the order.

The two conventions reverse the order relation. In standard notation for forcing, $q\le p$ means that $q$ is stronger than $p$. Thus $p$ and $q$ are incompatible forcing conditions when there is no $r$ with $r\le p$ and $r\le q$, while $D\subseteq \mathbb{P}$ is a dense set whenIn Jerusalem notation for forcing, $p\le q$ means that $q$ is stronger than $p$. Incompatibility therefore means that there is no $r$ with $p\le r$ and $q\le r$, and density means

The principal filter in an ordered set generated by $p$ contains exactly the elements above $p$. It is upward closed, and $p$ is a common lower bound in the filter for any two of its members. With standard notation for forcing, these are the conditions weaker than $p$. The principal filter need not be a generic filter: a dense set can require a strict strengthening. The compatible-condition filter in a forcing atom determines a ground-model generic filter includes strengthenings as well as weakenings and can therefore be strictly larger.

In standard notation for forcing, an order is separative if $p\nleqq q$ implies that some $r\le p$ is incompatible with $q$. Separativity does not imply an atomless forcing order: the one-point order is separative and has a forcing atom. Its only generic filter is a ground-model set. Atomlessness is the hypothesis in generic filter for an atomless order is new.