Let $T$ be a theory and let $\phi,\psi$ be statements that are independent of $T$. Say that $\psi$ is a $T$-weakening of $\phi$ if $T$ proves $\phi \Rightarrow \psi$ but cannot prove $\psi \Rightarrow \phi$, and say that $\phi$ is $T$-basic if there is no $T$-weakening of $\phi$.
If $T$ is at least as strong as Peano arithmetic, do $T$-basic sentences always exist? Is $\phi$ $T$-basic when $T$ is ZFC minus infinity and $\phi$ is the axiom of infinity?