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单词 IteratedForcing
释义

iterated forcing


We can define an iterated forcing of length α by inductionMathworldPlanetmath as follows:

Let P0=.

Let Q^0 be a forcingMathworldPlanetmath notion.

For βα, Pβ is the set of all functions f such that dom(f)β and for any idom(f), f(i) is a Pi-name for a member of Q^i. Order Pβ by the rule fg iff dom(g)dom(f) and for any idom(f), gif(i)Q^ig(i). (Translated, this means that any generic subset including g restricted to i forces that f(i), an element of Q^i, be less than g(i).)

For β<α, Q^β is a forcing notion in Pβ (so PβQ^β is a forcing notion).

Then the sequence Q^ββ<α is an iterated forcing.

If Pβ is restricted to finite functions that it is called a finite support iterated forcing (FS), if Pβ is restricted to countableMathworldPlanetmath functions, it is called a countable support iterated function (CS), and in general if each function in each Pβ has size less than κ then it is a <κ-support iterated forcing.

Typically we construct the sequence of Q^β’s by induction, using a function F such that F(Q^ββ<γ)=Q^γ.

Titleiterated forcing
Canonical nameIteratedForcing
Date of creation2013-03-22 12:54:47
Last modified on2013-03-22 12:54:47
OwnerHenry (455)
Last modified byHenry (455)
Numerical id5
AuthorHenry (455)
Entry typeDefinition
Classificationmsc 03E35
Classificationmsc 03E40
DefinesFS
DefinesCS
Definesfinite support
Definesfinite support iterated forcing
Definescountable support
Definescountable support iterated forcing
Definessupport iterated forcing
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更新时间:2025/5/4 10:16:04