Franklin's Notes

Generic extension model of ZFC

Let $\mathrm{M}$ be a countable transitive model of ZFC, and $P$ be a forcing notion of $\mathrm{M}$. Given a generic ideal $G\subset P$, the generic extension model $\mathrm{M}[G]$ is defined using the P-name hierarchy as follows: We may verify quickly that $\mathrm{M}\subset \mathrm{M}[G]$ and $G\in\mathrm{M}[G]$ as desired, by noticing that a P-name $\tau_x$ can be recursively defined for each $x\in\mathrm{M}$ as follows: and a P-name $\Gamma$ for $G$ can be defined as so that $\Gamma^G = G$ and $\tau_x^G = x$, as needed.

Verifying that $\mathrm{M}[G]$ satisfies all of the axioms of ZFC is non-trivial.





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