User:RJGray/Sandbox99

Other techniques: Cantor and Zilber
In the 1870s Georg Cantor started to develop set theory and in 1874 published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable. Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result. While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number, the proofs in both the aforementioned papers give methods to construct transcendental numbers.

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use of model theory in attempts to prove an unsolved problem in transcendental number theory. The problem is to determine the transcendence degree of the field
 * $$K=\mathbb{Q}(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})$$

for complex numbers x1,...,xn that are linearly independent over the rational numbers. Stephen Schanuel conjectured that the answer is n, but no proof is known. In 2004, though, Boris Zilber published a paper that used model theoretic techniques to create a structure that behaves very much like the complex numbers equipped with the operations of addition, multiplication, and exponentiation. Moreover, in this abstract structure Schanuel's conjecture does indeed hold. Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned, it could be that Schanuel's conjecture is false and that there exists some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture holds. Zilber did provide several criteria that would prove the structure in question was C, but could not prove the so-called Strong Exponential Closure axiom. The simplest case of this axiom has since been proved, but a proof that it holds in full generality is required to complete the proof of the conjecture.