Of course this has been explored: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Systems_which_contain_arithmetic https://en.wikipedia.org

Of course this has been explored: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Systems_which_contain_arithmetic https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... The bar is rather low, however (like Robinson arithmetic). Basically, you’d have to forgo integer arithmetics with multiplication.