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A remark on Zilber's pseudoexponentiation

Published online by Cambridge University Press:  12 March 2014

David Marker
David Marker*
Affiliation:
University of Illinois At Chicago, 851 S. Morgan Street Chicago, IL 60607-7045, USA.E-mail:marker@math.uic.edu
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Extract

When studying the model theory of

the first observation is that the integers can be defined as

Since ∂exp is subject to all of Gödel's phenomena, this is often also the last observation. After Wilkie proved that ℝexp is model complete, one could ask the same question for ∂exp, but the answer is negative.

Proposition 1.1. ∂expis not model complete

Proof. If ∂exp is model complete, then every definable set is a projection of a closed set. Since ∂ is locally compact, every definable set is Fσ. The same is true for the complement, so every definable set is also Gδ. But, since ℤ is definable, ℚ is definable and a standard corollary of the Baire Category Theorem tells us that ℚ is not Gδ.

Still, there are several interesting open questions about ∂exp.

• Is ℝ definable in ∂exp?

• (quasiminimality) Is every definable set countable or co-countable? (Note that this is true in the structure (∂, ℤ, +, ·) where we add a predicate for ℤ).

• (Mycielski) Is there an automorphism of ∂exp other than the identity and complex conjugation?1

A positive answer to the first question would tell us that ∂exp is essentially second order arithmetic, while a positive answer to the second would say that integers are really the only obstruction to a reasonable theory of definable sets.

A fascinating, novel approach to ∂exp is provided by Zilber's [6] pseudoexponentiation. Let L be the language {+, · E, 0, 1}.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 791 - 798
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Ax, J., On Schanuel's conjectures, Annals of Mathematics, vol. 2 (1971), no. 93, pp. 252–268.CrossRefGoogle Scholar
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[6]Zilber, B., Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2004), no. 1, pp. 67–95.CrossRefGoogle Scholar