I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. I want *natural statements*, e.g. statements that have been studied in number theory or combinatorics for their own sake. The motivation comes from looking for true statements that are not provable in $\mathsf{I}\Delta_0(L)$ where $L$ contains arbitrary fast growing (computable) functions.