Forcing partial orders and generic filters

Use q≤p to mean that q is stronger than p. In Cohen forcing P=Fn(ω,2), conditions are finite partial functions from natural numbers to zero or one, and the order is reverse inclusion. Two conditions are compatible when they have a common stronger extension, equivalently when they agree on their common domain. Their union then supplies such an extension. The empty function is a weakest condition. Two different values prescribed at the same coordinate are a genuine incompatibility, not a probability that can be averaged away.

A filter is nonempty, closed toward weaker conditions, and directed toward stronger common refinements. Thus if two pieces of information belong to the filter, some member of the filter refines both. Their union is consequently a function rather than a contradictory relation. A dense set D contains a refinement of every condition. Density is a condition-by-condition extension property; it does not mean most conditions in a counting or probabilistic sense. A single dense requirement may be met in many incompatible ways.

Given a nonempty partial order and a countable list of dense sets D_n, choose a descending sequence p_n meeting successive requirements. At each step density provides a stronger condition in the next dense set. The upward closure of this descending sequence is a filter meeting every D_n. This Rasiowa–Sikorski construction takes place in the surrounding mathematics, with its usual choice resources. A finite initial segment meets only finitely many requirements; the infinite construction is the part that supplies simultaneous genericity for the listed family.

Now assume externally that M is a countable transitive set model of ZFC and P∈M is a nonempty forcing partial order. Its dense subsets that are elements of M form an externally countable family, so the construction yields a P-generic filter over M. This is a conditional exposition, not a theorem that ZFC proves such an M exists. Completeness and Löwenheim–Skolem applied to consistency supply countable models, but do not make those models externally well-founded or transitive. General relative consistency proofs use additional formal metamathematical machinery rather than silently strengthening the model hypothesis.

For each n, the set of conditions specifying the nth coordinate is dense. A generic filter therefore has a union c defined on all of ω. For each real r∈2^ω∩M, the conditions disagreeing with r somewhere also form a dense set in M: extend any finite condition at a fresh coordinate using the opposite bit. Meeting that set guarantees c≠r. Thus the Cohen generic real is not a member of M, even though every finite part used to describe it belongs to M.

Meeting only the coordinate-domain requirements would not prove newness, because an old total real also meets all of them. The full ground-model dense family is essential. External countability does not conflict with M believing that it has uncountably many reals: the external enumeration need not be an element of M. Nor does genericity mean random sampling. The forcing construction specifies dense-set obligations and compatibility, and no assertion about a probability-one event is needed for the argument above.