TOPICS
Search

Wilkie's Theorem


Let phi(x_1,...,x_m) be an L_(exp) formula, where L_(exp)=L union {e^x} and L is the language of ordered rings L={+,-,·,<,0,1}. Then there exist n>=m and f_1,...,f_s in Z[x_1,...,x_n,e^(x_1),...,e^(x_n)] such that phi(x_1,...,x_n) is equivalent to

  exists x_(m+1)... exists x_nf_1(x_1,...,x_n,e^(x_1),...,e^(x_n))=... 
 =f_s(x_1,...,x_n,e^(x_1),...,e^(x_n))=0

(Marker 1996, Wilkie 1996). In other words, every formula is equivalent to an existential formula and every definable set is the projection of an exponential variety (Marker 1996).


Explore with Wolfram|Alpha

References

Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753-759, 1996.Wilkie, A. J. "Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function." J. Amer. Math. Soc. 9, 1051-1094, 1996.

Referenced on Wolfram|Alpha

Wilkie's Theorem

Cite this as:

Weisstein, Eric W. "Wilkie's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WilkiesTheorem.html

Subject classifications