iterated forcing and composition

There is a function satisfying forcingsMathworldPlanetmath are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if one is dense in the other f:Pα*QαPα+1.


Let f(g,q^)=g{α,q^}. This is obviously a member of Pα+1, since it is a partial functionMathworldPlanetmath from α+1 (and if the domain of g is less than α then so is the domain of f(g,q^)), if i<α then obviously f(g,q^) applied to i satisfies the definition of iterated forcing (since g does), and if i=α then the definition is satisfied since q^ is a name in Pi for a member of Qi.

f is order preserving, since if g1,q^1g2,q^2, all the appropriate characteristics of a function carry over to the image, and g1αPiq^1q^2 (by the definition of in *).

If g1,q^1 and g2,q^2 are incomparable then either g1 and g2 are incomparable, in which case whatever prevents them from being compared applies to their images as well, or q^1 and q^2 aren’t compared appropriately, in which case again this prevents the images from being compared.

Finally, let g be any element of Pα+1. Then gαPα. If αdom(g) then this is just g, and f(g,q^)g for any q^. If αdom(g) then f(gα,g(α))=g. Hence f[Pα*Qα] is dense in Pα+1, and so these are equivalent.

Title iterated forcing and composition
Canonical name IteratedForcingAndComposition
Date of creation 2013-03-22 12:54:51
Last modified on 2013-03-22 12:54:51
Owner Henry (455)
Last modified by Henry (455)
Numerical id 6
Author Henry (455)
Entry type Result
Classification msc 03E35
Classification msc 03E40