We study abstract versions of Goedela**s second incompleteness theorem and formulate generalizations of Loeba**s derivability conditions that work for logics weaker than the classical one. We isolate the role of the contraction and weakening rules in Goedela**s theorem and give a (toy) example of a system based on modal logic without contraction invalidating Goedela**s argument. On the other hand, Goedela**s theorem is established for a version of Peano arithmetic without contraction. (Joint work with Daniyar Shamkanov.)