Levy collapse

Given any cardinals κ and λ in 𝔐, we can use the Levy collapse to give a new model 𝔐[G] where λ=κ. Let P=Levy(κ,λ) be the set of partial functionsMathworldPlanetmath f:κλ with |dom(f)|<κ. These functions each give partial information about a function F which collapses λ onto κ.

Given any generic subset G of P, 𝔐[G] has a set G, so let F=G. Each element of G is a partial function, and they are all compatible, so F is a function. dom(G)=κ since for each α<κ the set of fP such that αdom(f) is dense (given any function without α, it is trivial to add (α,0), giving a stronger function which includes α). Also range(G)=λ since the set of fP such that α<λ is in the range of f is again dense (the domain of each f is bounded, so if β is larger than any element of dom(f), f{(β,α)} is stronger than f and includes λ in its domain).

So F is a surjective function from κ to λ, and λ is collapsed in 𝔐[G]. In addition, |Levy(κ,λ)|=λ, so it satisfies the λ+ chain condition, and therefore λ+ is not collapsed, and becomes κ+ (since for any ordinalMathworldPlanetmathPlanetmath between λ and λ+ there is already a surjective function to it from λ).

We can generalize this by forcingMathworldPlanetmath with P=Levy(κ,<λ) with κ regularPlanetmathPlanetmath, the set of partial functions f:λ×κλ such that f(0,α)=0, |dom(f)|<κ and if α>0 then f(α,i)<α. In essence, this is the product of Levy(κ,η) for each η<λ.

In 𝔐[G], define F=G and Fα(β)=F(α,β). Each Fα is a function from κ to α, and by the same argument as above Fα is both total and surjectivePlanetmathPlanetmath. Moreover, it can be shown that P satisfies the λ chain condition, so λ does not collapse and λ=κ+.

Title Levy collapse
Canonical name LevyCollapse
Date of creation 2013-04-16 22:08:32
Last modified on 2013-04-16 22:08:32
Owner ratboy (4018)
Last modified by e1568582 (1000182)
Numerical id 9
Author ratboy (1000182)
Entry type Example
Classification msc 03E45