#forcing #definition #forcing_notion **Def.** For any ordinal θ, the collapse poset Coll(ω, θ) consists of the finite partial functions from ω to θ, ordered by inclusion. For any nonzero ordinal θ, forcing with this poset adds a function from ω onto θ, making it countable in the forcing extension. **Lemma (Folklore).** Suppose that θ is any infinite ordinal. 1. Up to forcing equivalence, Coll(ω, θ) is the unique forcing notion of size |θ| necessarily collapsing θ to ω. 2. Coll(ω, θ) absorbs every forcing notion of size |θ|. 3. Coll(ω, θ) ∗ Coll(ω, λ) is forcing equivalent to Coll(ω, max{θ, λ}). Proof in [[Hamkins, Leibman, Löwe - Structural connections between a forcing class and its modal logic]] Let $\theta$ regular, $\kappa<\theta$ infinite cardinal. $\mathrm{Coll}(\kappa,\theta)=\mathrm{Fun}([\kappa]^{<\kappa},\theta)=\{ f:a\to \theta \mid a\in [\kappa]^{<\kappa}\}$ Ordered by reverse inclusion. # Levy collapse ## Definition Let $\theta$ strongly inaccessible, $\kappa<\theta$ infinite cardinal. $\mathrm{Coll}(\kappa,<\theta)=\{ f:a\to \theta \mid a\in [\theta \times \kappa]^{<\kappa}\land \forall\alpha,\beta\in a(f(\alpha,\beta)<\alpha)\}$ Ordered by reverse inclusion. ## Properties - $\mathrm{Coll}(\kappa,<\theta)$ is lt;\kappa$-closed, so $\kappa$ is not collapsed. - $\mathrm{Coll}(\kappa,<\theta)$ has $\theta$-[[Chain conditions|c.c.]] so $\theta$ is not collapsed. - All cardinals in $(\kappa,\theta)$ collapsed. - Hence $\Vdash \theta=\kappa ^{+}$ ## Variation Let $\theta$ strongly inaccessible, $\kappa<\theta$ infinite cardinal. $\mathrm{Coll}(\kappa,[ \mu,\theta))=\{ f:a\to \theta \mid a\in \big[[\mu,\theta) \times \kappa\big]^{<\kappa}\land \forall\alpha,\beta\in a(f(\alpha,\beta)<\alpha)\}$ Ordered by reverse inclusion. **Proposition.** $\mathrm{Coll}(\kappa,<\theta)\equiv \mathrm{Coll}(\kappa,<\mu)\times\mathrm{Coll}(\kappa,[\mu,\theta))$