Axiom: is a conventional assumption that almost generally accepted as true. For example, demand curve slopes downward, individual maximizes their utility….
Assumptions: are any set of hypothesis used to derive the claim, propositions, theorem. Usually, the assumption can be tested in emprical papers where theoritical economics base on the assumption to reach conclusion.
Claim: is a small theoritical result.
Proposition: is usual theoritical result. This mean the conclusion you conclude from the model is not very big suprising. In case, the result of your conclusion is one of the core ideas that change the views in the field it is name Theorem.
Theorem:is a big theoritical result.
Corollary: is result that can be derived from another result easily (proposition, theorem, claim)
Lemma : is result needed to help to prove another result (claim,proposition,theorem)
Remark: is the point which you should pay attention but which is not the main result for the proove of your claim, proposition or theorem.
Source: My observation on many articles
