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1
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Strong initial segments of models of IΔ₀

100%
D'Aquino P., Knight J.
Fundamenta Mathematicae
|
2007
|
tom 195
|
nr 2
155-176
EN
McAloon showed that if 𝓐 is a nonstandard model of IΔ₀, then some initial segment of 𝓐 is a nonstandard model of PA. Sommer and D'Aquino characterized, in terms of the Wainer functions, the elements that can belong to such an initial segment. The characterization used work of Ketonen and Solovay, and Paris. Here we give conditions on a model 𝓐 of IΔ₀ guaranteeing that there is an n-elementary initial segment that is a nonstandard model of PA. We also characterize the elements that can be included.
2
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Real closed exponential fields

81%
D'Aquino P., Knight J., Kuhlmann S., Lange K.
Fundamenta Mathematicae
|
2012
|
tom 219
|
nr 2
163-190
EN
Ressayre considered real closed exponential fields and "exponential" integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction and then analyze the complexity. Ressayre's construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering ≺ on R. The construction is clearly constructible over these objects. Each step looks effective, but there may be many steps. We produce an example of an exponential field R with a residue field section k and a well ordering ≺ on R such that $D^{c}(R)$ is low and k and ≺ are Δ⁰₃, and Ressayre's construction cannot be completed in $L_{ω₁^{CK}}$.
3
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Schanuel Nullstellensatz for Zilber fields

81%
D'Aquino P., Macintyre A., Terzo G.
Fundamenta Mathematicae
|
2010
|
tom 207
|
nr 2
123-143
EN
We characterize the unsolvable exponential polynomials over the exponential fields introduced by Zilber, and deduce Picard's Little Theorem for such fields.
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