***H: Maybe I misunderstood your question. Acceleration of motion is gauged based on a frame of reference. A: Oh really? Thanks, I didn't know that. This is great, really high-level debate :((***
Okay I'm glad you understand it, then let me respond to your statement which got this going: "why is it that you can't possibly run any experiment to prove that the definition of acceleration does not properly describe the motion of bodies in space?"
Dv/dt isn't a universal truth. It doesn't describe the acceleration of particles in a vacuum with no mass (e.g., photons). Photons do not accelerate in a vacuum they just go the speed of light. In case of electrons they jump instantaneously when orbiting an atom, dv/dt doesn't apply there.
Around black holes the measurement of acceleration depends on where the object and observer are in relation to the event horizon. The outsider might see the object having no acceleration (dv/dt=0), but as an object approaches the black hole, acceleration increases (i.e., from the objects' perspective actually increases).
I'm sure there are more examples where dv/dt are not valid. I apologize for assuming that you wished to talk about reference frames, but I assumed that your intent was to show that acceleration presented a philosophical dilemma in terms of not having an absolute frame of reference. Having replied to your post I see that you might have tried to show that acceleration is true by definition (I wrote this after replying to Alex). I'm sorry.
Warm regards, Harv