# Levy collapse

Given any cardinals $\kappa$ and $\lambda$ in $\mathfrak{M}$, we can use the Levy collapse to give a new model $\mathfrak{M}[G]$ where $\lambda=\kappa$. Let $P=\operatorname{Levy}(\kappa,\lambda)$ be the set of partial functions $f:\kappa\rightarrow\lambda$ with $|\operatorname{dom}(f)|<\kappa$. These functions each give partial information about a function $F$ which collapses $\lambda$ onto $\kappa$.

Given any generic subset $G$ of $P$, $\mathfrak{M}[G]$ has a set $G$, so let $F=\bigcup G$. Each element of $G$ is a partial function, and they are all compatible, so $F$ is a function. $\operatorname{dom}(G)=\kappa$ since for each $\alpha<\kappa$ the set of $f\in P$ such that $\alpha\in\operatorname{dom}(f)$ is dense (given any function without $\alpha$, it is trivial to add $(\alpha,0)$, giving a stronger function which includes $\alpha$). Also $\operatorname{range}(G)=\lambda$ since the set of $f\in P$ such that $\alpha<\lambda$ is in the range of $f$ is again dense (the domain of each $f$ is bounded, so if $\beta$ is larger than any element of $\operatorname{dom}(f)$, $f\cup\{(\beta,\alpha)\}$ is stronger than $f$ and includes $\lambda$ in its domain).

So $F$ is a surjective function from $\kappa$ to $\lambda$, and $\lambda$ is collapsed in $\mathfrak{M}[G]$. In addition, $|\operatorname{Levy}(\kappa,\lambda)|=\lambda$, so it satisfies the $\lambda^{+}$ chain condition, and therefore $\lambda^{+}$ is not collapsed, and becomes $\kappa^{+}$ (since for any ordinal between $\lambda$ and $\lambda^{+}$ there is already a surjective function to it from $\lambda$).

We can generalize this by forcing with $P=\operatorname{Levy}(\kappa,<\lambda)$ with $\kappa$ regular, the set of partial functions $f:\lambda\times\kappa\rightarrow\lambda$ such that $f(0,\alpha)=0$, $|\operatorname{dom}(f)|<\kappa$ and if $\alpha>0$ then $f(\alpha,i)<\alpha$. In essence, this is the product of $\operatorname{Levy}(\kappa,\eta)$ for each $\eta<\lambda$.

In $\mathfrak{M}[G]$, define $F=\bigcup G$ and $F_{\alpha}(\beta)=F(\alpha,\beta)$. Each $F_{\alpha}$ is a function from $\kappa$ to $\alpha$, and by the same argument as above $F_{\alpha}$ is both total and surjective. Moreover, it can be shown that $P$ satisfies the $\lambda$ chain condition, so $\lambda$ does not collapse and $\lambda=\kappa^{+}$.

Title Levy collapse LevyCollapse 2013-04-16 22:08:32 2013-04-16 22:08:32 ratboy (4018) e1568582 (1000182) 9 ratboy (1000182) Example msc 03E45