Wilkie,s Theorem
المؤلف:
Marker, D
المصدر:
"Model Theory and Exponentiation." Not. Amer. Math. Soc. 43
الجزء والصفحة:
...
5-1-2022
1302
Wilkie's Theorem
Let 
be an 
formula, where ![L_(exp)=L union <span style=]()
{e^x}" src="https://mathworld.wolfram.com/images/equations/WilkiesTheorem/Inline3.gif" style="height:18px; width:90px" /> and 
is the language of ordered rings ![L=<span style=]()
{+,-,·,<,0,1}" src="https://mathworld.wolfram.com/images/equations/WilkiesTheorem/Inline5.gif" style="height:15px; width:134px" />. Then there exist 
and ![f_1,...,f_s in Z[x_1,...,x_n,e^(x_1),...,e^(x_n)]](https://mathworld.wolfram.com/images/equations/WilkiesTheorem/Inline7.gif)
such that 
is equivalent to
(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).
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.
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