proof of partial order with chain condition does not collapse cardinals


Given any function f purporting to violate the theorem by being surjectivePlanetmathPlanetmath (or cofinal) on λ, we show that there are fewer than κ possible values of f(α), and therefore only max(α,κ) possible elements in the entire range of f, so f is not surjective (or cofinal).


Suppose λ>κ is a cardinal of 𝔐 that is not a cardinal in 𝔐[G].

There is some function f𝔐[G] and some cardinal α<λ such that f:αλ is surjective. This has a name, f^. For each β<α, consider

Fβ={γ<λpf^(β)=γ} for some pP

|Fβ|<κ, since any two pP which force different values for f^(β) are incompatible and P has no sets of incompatible elements of size κ.

Notice that Fβ is definable in 𝔐. Then the range of f must be contained in F=i<αFi. But |F|ακ=max(α,κ)<λ. So f cannot possibly be surjective, and therefore λ is not collapsed.

Now suppose that for some αλ>κ, cf(α)=λ in 𝔐 and for some η<λ there is a cofinal function f:ηα.

We can construct Fβ as above, and again the range of f is contained in F=i<ηFi. But then |range(f)||F|ηκ<λ. So there is some γ<α such that f(β)<γ for any β<η, and therefore f is not cofinal in α.

Title proof of partial order with chain condition does not collapse cardinals
Canonical name ProofOfPartialOrderWithChainConditionDoesNotCollapseCardinals
Date of creation 2013-03-22 12:53:43
Last modified on 2013-03-22 12:53:43
Owner Henry (455)
Last modified by Henry (455)
Numerical id 4
Author Henry (455)
Entry type Proof
Classification msc 03E35