Mistakes in Arnold? It's likely to have typos in any book, but real mistakes in Arnold? I doubt it.
Concerning the question about "Newton's axioms", one should know the history of Greiner's textbooks. They are meant to be used by students who start the general theory course already in the 1st semester, which was a novum at German universities at the time the books are written. I remember too well, how disappointed I was when I started to study physics at the university that theoretical physics was supposed to start in the 3rd semester only. But that were the good old times before the socalled "Bologna reform" on European universities, so you were free to do whatever you want, you only had to pass the "Vordiplom" exams after 2 years and the "Diplom" exams at the end. How you got the knowledge to do so nobody cared. So I went in the 3rd-semester theory lectures as a freshman. Greiner's books were among the best sources to read about all the many things I couldn't understand lacking 2 semesters preparation for theory through math and experimental physics lectures (which of course I also attended ;-)).
In such a situation you have to use an "inductive" approach, often using the "historical approach", i.e., to follow more or less the development of the subject as it was in history, which of course has to be cleaned up by all the thorny failures in getting to the modern understanding.
After having learned the (theoretical) physics in the inductive approach, it's also good to have a deductive point of view, i.e., you start from some real "axioms" and develop the phenomenology from that. Then you can state the Newtonian postulates (I'd no call them axioms) simply by defining the spacetime model to be the Galilei-Newton spacetime, which is a bundle of Euclidean spaces along a directed 1D time continuum, leading to the notion of Newton's "absolute space" as being defined by the equivalence classes of inertial frames, which exist by assumption. In the taste of 20th century physics, based on 19th century geometric views culminated in Klein's Erlanger Programm, you can (and in my opinion) should analyze it using group theory and symmetry analysis.
So why is the lex quarta valid: Simply because it is hypothesized within the choice of a spacetime model! Now one can think, how to interpret Newton's famous saying: "Hypothesis non fingo" ;-))).
