TITLE: Defining 'free monoid' without Nat?
QUESTION [5 upvotes]: Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined?  The definitions that I have been able to track down all use the natural numbers (since sequences/words need them).
In other words, I am trying to define the theory of free monoids (as a signature with sorts, operations and axioms), and I would like to know if I can do this without having the theory of the naturals already defined.  For exhibiting/constructing a free monoid, I do expect to need Nat.
Note that my ambient logic is higher-order, so I am fine with a second-order axiomatization.  If dependent-types are needed, that would also be acceptable.

REPLY [3 votes]: I am not this is what you want, but there is an old theorem of Levi that a monoid is free iff 

It is cancellative.
It has only trivial group of units.
It is equidivisible (you can google the definition; the key thing is that it is first order).
There are only finitely many principal right ideals containing any given principal right ideal.

1-3 are first order.  Probably 4 is a problem for what you want.