composition preserves chain condition

Let κ be a regular cardinal. Let P be a forcingMathworldPlanetmath notion satisfying the κ chain condition. Let Q^ be a P-name such that PQ^ is a forcing notion satisfying the κ chain condition. Then P*Q satisfies the κ chain condition.



We prove that there is some p such that any generic subset of P including p also includes κ of the pi. Then, since Q[G] satisfies the κ chain condition, two of the corresponding q^i must be compatible. Then, since G is directed, there is some p stronger than any of these which forces this to be true, and therefore makes two elements of S compatible.

Let S=pi,q^ii<κP*Q.

Claim: There is some pP such that p|{ipiG^}|=κ

(Note: G^={p,ppP}, hence G^[G]=G)

If no p forces this then every p forces that it is not true, and therefore P|{ipiG}|κ. Since κ is regular, this means that for any generic GP, {ipiG} is bounded. For each G, let f(G) be the least α such that β<α implies that there is some γ>β such that pγG. Define B={αα=f(G)} for some G.

Claim: |B|<κ

If αB then there is some pαP such that pf(G^)=α, and if α,βB then pα must be incompatible with pβ. Since P satisfies the κ chain condition, it follows that |B|<κ.

Since κ is regular, α=sub(B)<κ. But obviously pα+1pα+1G^. This is a contradictionMathworldPlanetmathPlanetmath, so we conclude that there must be some p such that p|{ipiG^}|=κ.

If GP is any generic subset containing p then A={q^i[G]piG} must have cardinality κ. Since Q[G] satisfies the κ chain condition, there exist i,j<κ such that pi,pjG and there is some q^[G]Q[G] such that q^[G]q^i[G],q^j[G]. Then since G is directed, there is some pG such that ppi,pj,p and pq^[G]q^1[G],q^2[G]. So p,q^pi,q^i,pj,q^j.

Title composition preserves chain condition
Canonical name CompositionPreservesChainCondition
Date of creation 2013-03-22 12:54:40
Last modified on 2013-03-22 12:54:40
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Result
Classification msc 03E40
Classification msc 03E35