Mathematical logic of the Galilean equation x' = x-vt challenged...
where x = distance and (x,y,z) = a point
Given Cartesian coordinate system S(x,y,z),
the abscissa x mathematically means the lineal distance along the x axis from S(0,0,0) to S(x,0,0).
Given Cartesian coordinate system S'(x',y',z'),
the abscissa x' mathematically means the lineal distance along the x' axis from S'(0,0,0) to S'(x',0,0).
Given that the abscissa x of S equals the abscissa x' of S'; [x' = x] when both systems are coincident, then it is true,
that should either system be moved from that coincidence a distance equal to vt,
that the abscissa of S = abscissa of S'; [x' = x],
therefore [the Galilean transformation equation abscissa of x' = abscissa of x -vt is incorrect]. Noting too that S'(x',y',z') = S(x-vt,y,z) is true.
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