#forcing_axioms #choice_principles [Useful axioms](<file:///home/ur/Dropbox/Papers/Viale - useful axioms.pdf>) # 1. The axiom of choice and Baire’s category theorem as forcing axioms ### Baire category theorem $\mathsf{BCT}_0\equiv$ For all compact Hausdorff spaces $(X,\tau)$ and all countable families $\{A_n:n\in\mathbb{N}\}$ of dense open subsets of $X$, $\bigcap_{n\in\mathbb{N}}A_n$ is non-empty. $\mathsf{BCT}_1\equiv$ Let $(P,\leq)$ be a partial order and $\{D_n:n\in\mathbb{N}\}$ be a family of predense subsets of $P$. Then there is a filter $G\subseteq P$ meeting all the sets $D_n$. $\mathsf{BCT}_1$ follows from some choice, perhaps [[Choice principles#Dependent choice|DC]], surely [[Choice principles|countable choice]] $\mathsf{BCT}_1 \implies \mathsf{BCT}_0$ $\mathsf{BCT}_0 \implies \mathsf{BCT}_1$ over [[Choice principles|prime ideal theorem]] ### Forcing axioms Let $\kappa$ be a cardinal and $(P,\leq)$ be a partial order. $FA_\kappa(P)\equiv$ For all families $\{D_\alpha:\alpha<\kappa\}$ of predense subsets of $P$, there is a filter $G$ on $P$ meeting all these predense sets. Given a class $\Gamma$ of partial orders $FA_\kappa(\Gamma)$ holds if $FA_\kappa(P)$ holds for all $P\in \Gamma$. $\Omega_\lambda$ denotes the class of posets $P$ for which $FA_\lambda(P)$ holds. $\Gamma_\lambda$ denotes the class of lt;\lambda$-closed posets. $FA_{\kappa}(\Gamma_{\kappa})$ means that for every lt;\kappa$-closed poset $P$ and every family $\{D_\alpha:\alpha<\kappa\}$ of predense subsets of $P$, there is a filter $G$ on $P$ meeting all these predense sets. **Theorem 1.8** $\mathsf{DC}_\kappa$ is equivalent to $FA_{\kappa}(\Gamma_{\kappa})$ over the theory $\mathsf{ZF}+\forall\lambda<\kappa\,\mathsf{DC}_\lambda$. Since $\mathsf{AC}$ is equivalent to $\forall\kappa\, \mathsf{DC}_\kappa$ modulo $\mathrm{ZF}$, we have that $\mathsf{AC}$ is equivalent to $\forall \kappa FA_{\kappa}(\Gamma_{\kappa})$ over $\mathrm{ZF}$ # 2. Large cardinals as forcing axioms **Def.** $\kappa$ is a measurable cardinal if and only if 1. there is a uniform lt;\kappa$-complete ultrafilter on the boolean algebra $\mathcal{P}(\kappa)$ 2. there is an ultrafilter $G$ on $\mathcal{P}(\kappa)/I$ (where $I$ is the ideal of bounded subsets of $\kappa$) which meets all the maximal antichains on $\mathcal{P}(\kappa)/I$ of size less than $\kappa$.\begin{definition} **Def.** Let $(P,\leq)$ be a partial order and $\mathcal{D}$ be a family of non-empty subsets of $P$. Let $\phi(x,y)$ be a property. - A filter $G$ on $P$ is $\mathcal{D}$-generic if $G\cap D$ is non-empty for all $D\in\mathcal{D}$. - $FA_\phi(P)$ holds if for any family $\mathcal{D}$ of predense subsets of $P$ such that $\phi(P,\mathcal{D})$ holds there is some $\mathcal{D}$-generic filter $G$ on $P$. # Boolean valued models, Los theorem, and generic absoluteness