List of Forcing Notions - Levy Collapsing

Levy Collapsing

These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.

• Collapsing a cardinal to ω: P is the set of all finite sequences of ordinals less than a given cardinal λ. If λ is uncountable then forcing with this poset collapses λ to ω.
• Collapsing a cardinal to another: P is the set of all functions from a subset of κ of cardinality less than κ to λ (for fixed cardinals κ and λ). Forcing with this poset collapses λ down to κ.
• Levy collapsing: If κ is regular and λ is inaccessible, then P is the set of functions p on subsets of λ× κ with domain of size less than κ and p(α,ξ)<α for every (α,ξ) in the domain of p. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.

Levy collapsing is named for Azriel Levy.