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We define a non-inertial reference as a reference frame which is accelerating.

But the question is if it is the reference frame itself what is it accelerating with respect to?

For simplification imagine that the this is the only reference frame in the universe and nothing exists outside, no reference frame outside. If this frame is accelerating what is it accelerating with respect to?

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No, we defined an inertia frame as a frame in which Newton's laws of motion (in particular, the third law) hold. And then define a non-inertial frame as one which is accelerating w.r.t. inertial frames. In a non-inertial frame, there will be pseudo forces with no reactions, and in other words, Newton's laws of motion do not hold.

Newton's first law state that there are inertial frames in this universe. So your worries do not hold.

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  • $\begingroup$ To be more precise, we defined an inertia frame as a frame in which the first of Newton's law holds. $\endgroup$ – gented Dec 12 '16 at 14:31
  • $\begingroup$ Newton's law states , but there aren't any perfect inertial frames. Are there any ? physics.stackexchange.com/q/264128 $\endgroup$ – Shashaank Dec 12 '16 at 18:47
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In a 3-dimensional general relativity sense, an inertial reference frame is one in which a mass attached to a set of 3 mutually-perpendicular springs does not distort the springs. That definition treats systems with gravity as non-inertial, which is difficult for most non-professional physicists, but necessary for astrophysicists and cosmologists.

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