Definition of Newton's first law I have always had a doubt in the definition of the Newton's first law. In general, it is stated in a form like:

An object at rest remains at rest, or if in motion, remains in motion at a constant velocity unless acted on by a net external force.

However, we know that there are reference frames in which the first law is not valid, these are the called non-inertial reference frames. So the first law should be stated as "there are reference frames in which an object at rest remains at rest, or if in motion, remains in motion at a constant velocity unless acted on by a net external force". But the most books don't care about this and state the first law like the first way, what to me is incorrect.
I understand a high school book doesn't talk about non-inertial reference frames (this is not simple), but I have already seen a lot of undergraduate and graduate physics books do this. What do you think? This statement is really incorrect? 
 A: Many texts are encumbered by too much tradition in many ways when it comes to explaining important, basic concepts.
The "proper" statement of Newton's first law should have two parts. One of these is the definition of an inertial frame of reference: this is a frame of reference in which all objects which are not being acted on by any forces, i.e. are not interacting with other objects, will move with steady motion. The second part is that the ways in which objects move and interactions behave are such that it is possible to have such a frame.
The last part is a physical law, because we can imagine a world where it does not hold, but we cannot imagine a world where a "definition" doesn't hold "physically" since definitions are statements of what words mean and that is something purely in our heads (We could imagine though, of course, a world where people use the word differently and thus don't accept such a definition, but not the definition itself).
A world where the "law" part of Newton's first law doesn't hold is a world where no inertial frames exist, i.e. nothing in it satisfies the definition, but that's not the same as the definition being wrong (e.g. I could define a "zneezax" as something that is "a piece of candy-like dragon blood that glows bright pink". No zneezaxes exist, as far as we know, but that doesn't invalidate the definition).
In fact, however, it is quite difficult to imagine such a world, but not impossible. Because, it turns out, if you have a bunch of objects in fixed paths of motion you can,  with rather clever and weird choices of complicated, curvilinear coordinate systems that morph over time (if you don't like that last part, keep in mind that a simply-moving system is a simple form of such "morphing"), no matter how they're moving, make them all "at rest" or in "steady motion", i.e. that their coordinates do not change. To rule that out, you actually need to thus quantify over an unlimited number of possibilites including ones counterfactual to the actual situation at hand.
(For a simple concrete example, consider a "universe" with different and very simple laws of physics in which its sole content is two separate, non-interacting, point-like objects, that oscillate forever, back and forth with respect to each other, purely on their own, with no connection between them. Define a coordinate system that compresses and rarefies in the direction of their oscillation accordingly. Now they are steady with regard thereto. It would only be by imposing a third, counterfactual object, that one could regard this coordinate system as not inertial.)
Hence, I think a better statement may be, after some thought on this:

"It is possible to impose upon the space and time of the Universe a coordinate system for which, with regard to the same system in all cases, an arbitrary number of arbitrarily-configured objects, were such to exist, would move in a steady fashion, unless some are in interaction with each other, in which case, only those objects who are not in such interaction, will be assured to move steadily."

then to follow it with:

"Such coordinate systems, are what we call inertial systems."

The first is the law part, the second part is the definition of the inertial system. As one can see, this is quite clearly a law since it requires that the laws governing the motion and interaction of objects are such as to make this possible.
A: Newton starts with the assumption that there is a special preferred frame of reference:
Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external...



Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. 

It's clear from context that Newton intends his laws to refer to measurements in that "absolute" frame.  So Newton, at least, does not have to qualify his laws further at this point because he's already qualified them up front by specifying the frame he's working in.
I don't know what books you're talking about, but if they're following Newton, then perhaps they're doing exactly the same thing.
A: The general form of Newton's first law itself turns out to be the definition of inertial frame of reference to some extent.
It is not exactly a definition but a law. Looking it in another way it leads to the definition of inertial frame.
