I never went deep into formal logic, so I can misname a few things as well. 1. When it comes to the Gödel's incompleteness theorem - for non-recursive systems

I never went deep into formal logic, so I can misname a few things as well. 1. When it comes to the Gödel's incompleteness theorem - for non-recursive systems you may you can get without that, vide https://math.stackexchange.com/questions/3359878/what-is-some-easy-illustrative-example-of-an-non-recursive-formal-system-and-rec https://math.stackexchange.com/questions/3359878/what-is-som... 2. In general, you can do a lot without equivalence. For example, in set theory (ZF), you get only trivial equivalence (sets having the same elements are the same set). But you can still do a lot with "contains". A similar thing works with partially ordered sets, in which you have the "<=" operator.