A problem by David Mattingly
If light travels infinitely fast then Finn's clock will not tick between the sending and receiving of the light. Therefore \(F_1=A_r=F_f\). This means that every observer has the exact same tick value on their synchronized clocks - i.e. there is a global time function \(t\) defined everywhere. No matter where various observers are, on the earth, moon, Jupiter, etc. they can all set their clocks to tick together. We define time as the single, unique global value of the ticking clocks. (Remember, this is in Newtonian mechanics).
This physical assumption about how light travels is reflected in the basic structure of Newtonian mechanics, although you may never have realized it. Consider the Galilean transformations that relate the \(x,t\) coordinates of one observer (Finn) with the \(x',t'\) coordinates of another observer (Aaron) moving at a relative velocity. Do the Galilean transformations change the time used by the two observers? i.e. is \(t'=t\) or not?