Editing Transcendental number theory (section)

===Other techniques: Cantor and Zilber===

In the 1870s, [[Georg Cantor]] started to develop [[set theory]] and, in 1874, published a [[Georg Cantor's first set theory article|paper]] proving that the algebraic numbers could be put in [[Bijection|one-to-one correspondence]] with the set of [[natural number]]s, and thus that the set of transcendental numbers must be [[uncountable set|uncountable]].<ref>{{cite journal |first=G. |last=Cantor |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 |title=Ueber eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=1874 |year=1874 |issue= 77|pages=258–262 |doi=10.1515/crll.1874.77.258 |s2cid=199545885 |language=de}}</ref>  Later, in 1891, Cantor used his more familiar [[Cantor's diagonal argument|diagonal argument]] to prove the same result.<ref>{{cite journal |first=G. |last=Cantor |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |year=1891 |pages=75–78 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002113910 |language=de}}</ref>  While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,<ref>{{cite book |first1=M. |last1=Kac |first2=U. |last2=Stanislaw |title=Mathematics and Logic |url=https://archive.org/details/mathematicslogic0000kacm |url-access=registration |publisher=Fredering A. Praeger |year=1968 |page=[https://archive.org/details/mathematicslogic0000kacm/page/13 13] }}</ref><ref>{{cite book |first=E. T. |last=Bell |title=Men of Mathematics |location=New York |publisher=Simon & Schuster |year=1937 |page=[https://archive.org/details/menofmathematics0041bell/page/569 569] |title-link=Men of Mathematics }}</ref> the proofs in both the aforementioned papers give methods to construct transcendental numbers.<ref>{{cite journal |first=R. |last=Gray |title=Georg Cantor and Transcendental Numbers |url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf |journal=[[American Mathematical Monthly]] |volume=101 |year=1994 |issue=9 |pages=819–832 |jstor=2975129 |doi=10.1080/00029890.1994.11997035 }}</ref>

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
:<math>K=\mathbb{Q}(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})</math>
for complex numbers ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that are linearly independent over the rational numbers.  [[Stephen Schanuel]] [[Schanuel's conjecture|conjectured]] that the answer is at least ''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 number]]s equipped with the operations of addition, multiplication, and exponentiation.  Moreover, in this abstract structure Schanuel's conjecture does indeed hold.<ref>{{cite journal |first=B. |last=Zilber |title=Pseudo-exponentiation on algebraically closed fields of characteristic zero |journal=Annals of Pure and Applied Logic |volume=132 |year=2005 |issue=1 |pages=67–95 |mr=2102856 |doi=10.1016/j.apal.2004.07.001 |doi-access=free }}</ref>  Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned; there could exist some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture doesn't hold.  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,<ref>{{cite journal |first=D. |last=Marker |title=A remark on Zilber's pseudoexponentiation |journal=Journal of Symbolic Logic |volume=71 |issue=3 |year=2006 |pages=791–798 |mr=2250821 |jstor=27588482 |doi=10.2178/jsl/1154698577|s2cid=1477361 }}</ref> but a proof that it holds in full generality is required to complete the proof of the conjecture.