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A family of $$ 2^{\aleph _1 } $$ logarithmic functions of distinct growth rates

100%

Kuhlmann S.

Open Mathematics

2010

tom 8

nr 6

1026-1028

We construct a totally ordered set Γ of positive infinite germs (i.e. germs of positive real-valued functions that tend to +∞), with order type being the lexicographic product ℵ1 × ℤ2. We show that Γ admits $$ 2^{\aleph _1 } $$ order preserving automorphisms of pairwise distinct growth rates.

Real closed exponential fields

51%

D'Aquino P., Knight J., Kuhlmann S., Lange K.

Fundamenta Mathematicae

2012

tom 219

nr 2

163-190

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 $L_{ω₁^{CK}}$.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Open Mathematics

2 Kuhlmann S.

1 D'Aquino P.

1 Knight J.

1 Lange K.

1 2012

1 2010

Wyniki wyszukiwania

A family of $$ 2^{\aleph _1 } $$ logarithmic functions of distinct growth rates

Real closed exponential fields