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"

Is there any 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"