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This is my first post here. I have a line of thoughts which might be wrong but I couldn't find the error.

Suppose we have a point mass subjected to an arbitrary force $F$. In the lab reference frame we see it accelerating. In the reference frame where the mass is at rest we see the mass at rest. In this frame we don't see any force so the first law of dynamics is respected. We must conclude that this is an inertial reference frame and therefore every time I put myself in such a reference frame I am always in an inertial one.

This is clearly wrong since the reference frame is accelerating with respect to the lab reference frame and therefore it can't be inertial.

Where's the flaw?

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    $\begingroup$ I'm not sure I understand the problem; to see the mass as stationary you must be accelerating yourself, which is not something you can just ignore. And once you account for it in then everything makes sense... $\endgroup$
    – lemon
    Commented Jun 21, 2016 at 9:26
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    $\begingroup$ Newton's laws are only valid in inertial systems. In the system with your accelerating mass you could simply let go of a small test mass and it would magically accelerate away from you (also without any external forces!). You would immediately know that you are not in a rest system. Just look around you... do you see any inertial system physics? I don't. Well... maybe you are living on the ISS, right now, and you do, but what are the chances that an astronaut on the ISS has such a question about inertial systems? $\endgroup$
    – CuriousOne
    Commented Jun 21, 2016 at 9:45
  • $\begingroup$ @tfb and CuriousOne so that's the only way to understand i'm not in an inertial frame? Performing an experiment? mental or practical. $\endgroup$
    – RenatoRenatoRenato
    Commented Jun 21, 2016 at 10:08
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    $\begingroup$ @Angeloferrari Yes: physics is an experimental science so you only find out anything by doing experiments (sorry if that seems patronising: it's not meant to be: rather it's a really important point which is easy to forget). $\endgroup$
    – user107153
    Commented Jun 21, 2016 at 10:37
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    $\begingroup$ @tfb well it doesn't sound patronising. Thank you for your help. I don't think you can define any science in a way different than experimental. It wouldn't be science if it weren't. Unless you use the word science in that broad meaning they use it nowadays which gives the status of sciences to things like social or religious sciences. $\endgroup$
    – RenatoRenatoRenato
    Commented Jun 21, 2016 at 10:59

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The flaw is that you've failed to do an experiment which will tell you whether the frame is inertial or not. If you do such an experiment -- for instance take a test mass, initially at rest with respect to the frame, release it, and see if it remains at rest -- you will immediately discover that the frame is not inertial.

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The flaw is your assumption that

In this [accelerating] frame we don't see any force so the first law of dynamics is respected.

In the accelerating reference frame you do see evidence of a force, even though you don't see the effect you are expecting (acceleration of the object).

Like an observer standing on the surface of the Earth, acted on by the Earth's gravitational field, the object experiences the push of a contact force - eg from the ground. Or if the object is being rotated in a circle on the end of a string, it experiences a pull from the string. If the object is elastic there will be some deformation (contraction, elongation, change of shape), which indicates that a force is acting.

As tbf and CuriousOne point out, you could also actively perform an experiment to test if you are in an inertial frame, rather than passively observe. EG release a test mass, or weigh it.

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