# iterated forcing and composition

There is a function satisfying forcings^{} are equivalent^{} if one is dense in the other $f:{P}_{\alpha}*{Q}_{\alpha}\to {P}_{\alpha +1}$.

## Proof

Let $f(\u27e8g,\widehat{q}\u27e9)=g\cup \{\u27e8\alpha ,\widehat{q}\u27e9\}$. This is obviously a member of ${P}_{\alpha +1}$, since it is a partial function^{} from $\alpha +1$ (and if the domain of $g$ is less than $\alpha $ then so is the domain of $f(\u27e8g,\widehat{q}\u27e9)$), if $$ then obviously $f(\u27e8g,\widehat{q}\u27e9)$ applied to $i$ satisfies the definition of iterated forcing (since $g$ does), and if $i=\alpha $ then the definition is satisfied since $\widehat{q}$ is a name in ${P}_{i}$ for a member of ${Q}_{i}$.

$f$ is order preserving, since if $\u27e8{g}_{1},{\widehat{q}}_{1}\u27e9\le \u27e8{g}_{2},{\widehat{q}}_{2}\u27e9$, all the appropriate characteristics of a function carry over to the image, and ${g}_{1}\upharpoonright \alpha {\u22a9}_{{P}_{i}}{\widehat{q}}_{1}\le {\widehat{q}}_{2}$ (by the definition of $\le $ in $*$).

If $\u27e8{g}_{1},{\widehat{q}}_{1}\u27e9$ and $\u27e8{g}_{2},{\widehat{q}}_{2}\u27e9$ are incomparable then either ${g}_{1}$ and ${g}_{2}$ are incomparable, in which case whatever prevents them from being compared applies to their images as well, or ${\widehat{q}}_{1}$ and ${\widehat{q}}_{2}$ aren’t compared appropriately, in which case again this prevents the images from being compared.

Finally, let $g$ be any element of ${P}_{\alpha +1}$. Then $g\upharpoonright \alpha \in {P}_{\alpha}$. If $\alpha \notin \mathrm{dom}(g)$ then this is just $g$, and $f(\u27e8g,\widehat{q}\u27e9)\le g$ for any $\widehat{q}$. If $\alpha \in \mathrm{dom}(g)$ then $f(\u27e8g\upharpoonright \alpha ,g(\alpha )\u27e9)=g$. Hence $f[{P}_{\alpha}*{Q}_{\alpha}]$ is dense in ${P}_{\alpha +1}$, 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 |