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

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Cohen forcing raises and, under a name count, fixes the continuum

Statement

In ZFC, $Add (ω, λ)$ for infinite $λ$ preserves cardinals and forces $2^{ℵ_{0}} \geq λ$ . If the ground model also satisfies $λ^{ℵ_{0}} = λ$ and $λ \geq 2^{ℵ_{0}}$ , then it forces $2^{ℵ_{0}} = λ$ . In particular, over GCH, $Add (ω, ℵ_{2})$ forces the continuum to be exactly $ℵ_{2}$ .

Facts & Assumptions

Given: AC and infinite $λ$ .

[F1]

Closure and chain conditions of Cohen forcing gives ccc.

[F2]

Chain conditions preserve high cofinalities and ccc preserves cardinals gives cardinal preservation.

[F3]

Cohen coordinates are distinct and mutually generic gives $λ$ distinct reals.

[F4]

Nice-name reduction and the ccc counting bound bounds real names.

[F5]

Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations supplies exponent calculations.

Proof

1.1F1F2F3

By F1, F2 all cardinals are preserved. F3 supplies an injection $λ ↪ P (ω)$ in the extension, so $2^{ℵ_{0}} \geq λ$ .

2.1F4F5step 1.1

The forcing has size $λ$ . Under the additional hypotheses F4 gives at most $λ^{ℵ_{0}} = λ$ nice names for reals. Every real has one, so $2^{ℵ_{0}} \leq λ$ ; combine with step 1.1.

3.1F2F5step 2.1∎

Under GCH, the ground continuum is $ℵ_{1}$ and $(ℵ_{2})^{ℵ_{0}} = ℵ_{2}$ . Apply step 2.1 with $λ = ℵ_{2}$ ; preservation ensures that this remains the extension's $ℵ_{2}$ . AC is used in F2, F4, and the cardinal computations.

Depends on

Dependency tree · 2 levels

Sources

Karagila, Forcing & Symmetric Extensions, Theorem 3.31 and Corollary 3.33