Generalisations
Wilkie proved that there are statements about the positive integers that cannot be proved using the eleven axioms above and showed what extra information is needed before such statements can be proved. Using Nevanlinna theory it has also been proved that if one restricts the kinds of exponential one takes then the above eleven axioms are sufficient to prove every true statement.
Another problem stemming from Wilkie's result that remains open is that which asks what the smallest algebra is for which W(x, y) is not true but the eleven axioms above are. In 1985 an algebra with 59 elements was found that satisfied the axioms but for which W(x, y) was false. Smaller such algebras have since been found, and it is now known that the smallest such one must have either 11 or 12 elements.
Read more about this topic: Tarski's High School Algebra Problem