# Axiom of choice $\prod_{i\in I}A_i$ is non-empty for all families of non empty sets $\{A_i:i\in I\}$, i.e. there is a choice function $f:I\to\bigcup_{i\in I} A_i$ such that $f(i)\in A_i$ for all $i\in I$. # Dependent choices Let $\kappa$ be an infinite cardinal. The *principle of dependent choices* $\mathsf{DC}_\kappa$ states the following: For every non-empty set $X$ and every function $F\colon X^{<\kappa}\to\mathcal{P}(X)\setminus\{\emptyset\}$, there exists $g\colon\kappa\to X$ such that $g(\alpha)\in F(g\restriction\alpha)$ for all $\alpha<\kappa$. Seems to me that this basically means that we can do definitions by recursion on $\kappa$. $\mathsf{AC}$ is equivalent to $\forall\kappa\, \mathsf{DC}_\kappa$ modulo $ZF$. # Prime ideal theorem Let _B_ be a Boolean algebra, let _I_ be an ideal and let _F_ be a filter of _B_, such that _I_ and _F_ are [disjoint](https://en.wikipedia.org/wiki/Disjoint_set "Disjoint set"). Then _I_ is contained in some prime ideal of _B_ that is disjoint from _F_.