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Wilkie,s Theorem

المؤلف:  Marker, D

المصدر:  "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43

الجزء والصفحة:  ...

5-1-2022

1511

Wilkie's Theorem

 

Let  be an  formula, where {e^x}" src="https://mathworld.wolfram.com/images/equations/WilkiesTheorem/Inline3.gif" style="height:18px; width:90px" /> and  is the language of ordered rings {+,-,·,<,0,1}" src="https://mathworld.wolfram.com/images/equations/WilkiesTheorem/Inline5.gif" style="height:15px; width:134px" />. Then there exist  and  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).

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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.