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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.

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