Where is the least upper bound property used in transcendence proofs? The second-order theory of real numbers is what you get when you take the axioms for ordered fields and add one more axiom, the least upper bond property, also known as Dedekind completeness: that every set that has an upper bound has a least upper bounds.  This is a theory in the language of second-order logic, and it's categorical, meaning it only has one model.  But if we make it into a first-order theory, taking only the first-order consequences of the completeness axiom, then the resulting theory is no longer categorical: in addition to the standard model, the real number systems, we get all sorts of non-standard models, known as real closed fields.
One example of a real closed field is the set of algebraic real numbers.  That means that the proofs that pi and e are transcendental numbers (and possibly even the definition of pi and e) cannot be carried out in the theory of real closed fields.  In other words, the proof of each must rely on some second-order consequence of the completeness axiom.  My question is, what is the second-order consequence that is used in the proof of each, and where in the proof is it used?
Any help would be greatly appreciated.
Thank You in Advance.
 A: If you look at the [wikipedia entry] (http://en.wikipedia.org/wiki/Real_closed_field) on real closed fields, you see that one characterization of such ordered fields is that the intermediate value theorem holds for polynomials over the field; related to this is the fact that
odd degree polynomials always have a root.  This gives a way of constructing
certain numbers in $F$, say as the root of a certain odd degree polynomial.
Since $\pi$ and $e$ are transcendental, we know they can't be constructed this way (as roots of polynomials over the field of real algebraic numbers).  And the other usual definitions of $\pi$ and $e$ are in terms of infinite series, or integrals, or other limiting processes which don't apply in a general real closed field.  
So the basic second order property that is being used is the existence of certain limits to define $\pi$ and $e$.   The ability to define numbers as limits is something that one has available in the reals that is not available in a general real ordered field.
