Solution
In 1980 Alex Wilkie proved that not every identity in question can be proved using the axioms above. He did this by explicitly finding such an identity. By introducing new function symbols corresponding to polynomials that map positive numbers to positive numbers he proved this identity, and showed that these functions together with the eleven axioms above were both sufficient and necessary to prove it. The identity in question is
This identity is usually denoted W(x,y) and is true for all positive integers x and y, as can be seen by factoring out of the second terms; yet it cannot be proved true using the eleven high school axioms.
Intuitively, the identity cannot be proved because the high school axioms can't be used to discuss the polynomial . Reasoning about that polynomial and the subterm requires a concept of negation or subtraction, and these are not present in the high school axioms. Lacking this, it is then impossible to use the axioms to manipulate the polynomial and prove true properties about it. Wilkie's results from his paper show, in more formal language, that the "only gap" in the high school axioms is the inability to manipulate polynomials with negative coefficients.
Read more about this topic: Tarski's High School Algebra Problem
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