This is what provides mathematics with its greatest strength - truth can be demonstrated, there is no ambiguity about a mathematical truth; and its greatest weakness - mathematics can not prove anything that is not mathematical.
Symbols: M, I, U
Propositions: Strings of these symbols. (may be empty)
Grammar: (Rules of Production) x is any proposition.
A theorem is a proposition which can be obtained from the axioms by applying the rules of the grammar.
For instance, MUIU is a theorem.
MI --(2) MII --(2) MIIII --(3) MUI --(1) MUIU.
MUIIU is also a theorem.
MI --(above) MUIU --(2) MUIUUIU --(4) MUIIU.
But clearly U is not a theorem.
Question: Is MU a theorem?