Is there  an recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is  deduced from the recursively axiomatic system in infinite ways.Could any one give an example or proof?

**EDIT**:We define a proof or a reduced proof as one which could not omit any proposition  in the arguments,otherwise would be invalid. Or we can view it as question of formal language,that is :is there a formal language with at least one word which can be parsed in infinitely many ways?Of course there have to be no rewriting like "irrelevent padding or stupid detours"