# iterated forcing

We can define an *iterated forcing* of length $\alpha $ by induction^{} as follows:

Let ${P}_{0}=\mathrm{\varnothing}$.

Let ${\widehat{Q}}_{0}$ be a forcing^{} notion.

For $\beta \le \alpha $, ${P}_{\beta}$ is the set of all functions $f$ such that $\mathrm{dom}(f)\subseteq \beta $ and for any $i\in \mathrm{dom}(f)$, $f(i)$ is a ${P}_{i}$-name for a member of ${\widehat{Q}}_{i}$. Order ${P}_{\beta}$ by the rule $f\le g$ iff $\mathrm{dom}(g)\subseteq \mathrm{dom}(f)$ and for any $i\in \mathrm{dom}(f)$, $g\upharpoonright i\u22a9f(i){\le}_{{\widehat{Q}}_{i}}g(i)$. (Translated, this means that any generic subset including $g$ restricted to $i$ forces that $f(i)$, an element of ${\widehat{Q}}_{i}$, be less than $g(i)$.)

For $$, ${\widehat{Q}}_{\beta}$ is a forcing notion in ${P}_{\beta}$ (so ${\u22a9}_{{P}_{\beta}}{\widehat{Q}}_{\beta}$ is a forcing notion).

Then the sequence $$ is an iterated forcing.

If ${P}_{\beta}$ is restricted to finite functions that it is called a *finite support iterated forcing* (FS), if ${P}_{\beta}$ is restricted to countable^{} functions, it is called a *countable support iterated function* (CS), and in general if each function in each ${P}_{\beta}$ has size less than $\kappa $ then it is a *$$-support iterated forcing*.

Typically we construct the sequence of ${\widehat{Q}}_{\beta}$’s by induction, using a function $F$ such that $$.

Title | iterated forcing |

Canonical name | IteratedForcing |

Date of creation | 2013-03-22 12:54:47 |

Last modified on | 2013-03-22 12:54:47 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 5 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E35 |

Classification | msc 03E40 |

Defines | FS |

Defines | CS |

Defines | finite support |

Defines | finite support iterated forcing |

Defines | countable support |

Defines | countable support iterated forcing |

Defines | support iterated forcing |