ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Two Cohen reals as mutually generic coordinates

Statement

Let $M$ be a transitive ZFC model and let $G$ be $M$ -generic for $Add (ω, 2)$ . Write $c_{i} (n) = (⋃ G) (i, n)$ for $i < 2$ . The forcing $Add (ω, 2)$ is isomorphic to $Add (ω, 1) \times Add (ω, 1)$ . Its coordinate reals satisfy $M [c_{0}, c_{1}] = M [c_{0}] [c_{1}] = M [c_{1}] [c_{0}]$ , and each is Cohen-generic over the extension by the other.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Cohen coordinates are distinct and mutually generic gives product genericity.

Proof

1.1

Map $p$ to $(r_{0}, r_{1})$ , where $r_{i} (0, n) = p (i, n)$ whenever $(i, n) \in dom p$ . Each $r_{i}$ is a condition in $Add (ω, 1)$ ; conversely recover $p$ from $(r_{0}, r_{1})$ by $p (i, n) = r_{i} (0, n)$ . These maps are inverse and preserve reverse inclusion and compatibility. F1 applied to the partition $2 = {0} \dot{\cup} {1}$ yields the three model equalities and mutual genericity.

2.1

For distinctness, below any pair of finite conditions choose a fresh $n$ and extend the first with bit $0$ and the second with bit $1$ . The resulting dense set is met, so the coordinate-union definition in F1 gives $c_{0} (n) \neq c_{1} (n)$ .

F1∎

Depends on

Cohen coordinates are distinct and mutually generic

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karagila, Forcing & Symmetric Extensions, product factorization (standard reference, not scraped)