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 order preserving automorphisms of pairwise distinct growth rates.
@article{bwmeta1.element.doi-10_2478_s11533-010-0070-z,
author = {Salma Kuhlmann},
title = {A family of \[ 2^{\aleph \_1 } \]
logarithmic functions of distinct growth rates},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {1026-1028},
zbl = {1214.03024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0070-z}
}
Salma Kuhlmann. A family of \[ 2^{\aleph _1 } \]
logarithmic functions of distinct growth rates. Open Mathematics, Tome 8 (2010) pp. 1026-1028. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0070-z/
[1] Hardy G.H., Orders of Infinity; The ‘Infinitärcalcül’ of Paul Du Bois-Reymond, Cambridge Tracts in Math. and Math. Physics, 12, Cambridge University Press, Cambridge, 1954
[2] Hausdorff F., Die Graduierung nach dem Endverlauf, Abhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig. Math.-Phys. Klasse, 1909, 31, 295–334 | Zbl 40.0446.02
[3] Kojman M., History of singular cardinals in the 20th century: from Hausdorff’s gap to Shelah’s PCF theory, Handb. Hist. Log., 6 (in press)
[4] Kuhlmann S., Ordered Exponential Fields, Fields Inst. Monogr., 12, American Mathematical Society, Providence, 2000 | Zbl 0989.12003
[5] Kuhlmann S., Shelah S., κ-bounded exponential-logarithmic power series fields, Ann. Pure Appl. Logic, 2005, 136(3), 284–296 http://dx.doi.org/10.1016/j.apal.2005.04.001 | Zbl 1079.03024
[6] Steprāns J., History of the continuum in the 20th century, Handb. Hist. Log., 6 (in press)