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please delete it
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The Universe doesn't come equipped with any "clocks at each moment" that would immediately recognize different "moments".
Indeed, the laws of physics - except for cosmology - are totally invariant with respect to translations in time, which is just a different way of saying that there is no way, even in principle, to find out whether an event occurred at time $t_1$ or $t_2$: the events are guaranteed to proceed in an identical way if the initial conditions are identical, whether the events occur at $t_1$ or $t_2$. By Noether's theorem, this symmetry is inseparably connected with energy conservation. Still, spacetime exists and events occur at different times, and the distances between events in time - duration of processes etc. - are well-defined numbers and can be measured by various tools.
In particular, the proper time of world lines may be measured by all kinds of clocks.
In the context of cosmology, the Universe is expanding and the time-translational symmetry is broken by the expansion (and so is the corresponding energy conservation law). It follows that one may define a "cosmic time" - the proper time of the "static" (a frame in which the density of momentum of the CMB radiation vanishes) world line stretched between a point of the Big Bang and a present event. This cosmic time is related to the current CMB temperature and many other things. Still, the CMB temperature is obviously not accurate enough to measure the time separations of some very recent events from each other. In that case, we prefer things like atomic clocks that may achieve an unbelievable precision.
Archeologists use radioactive isotopes to determine the age of various things because it works and they don't use a more accurate method because no method that would be more accurate is available. Pharaohs could have put (and reset) digital clocks into the pyramids when they built them except that they failed to do so and it's very hard to prosecute them for their negligence today. ;-) At any rate, those questions are for geologists, historians, and archeologists - not physicists. If a historian asked whether physicists may come up with a better method, the answer is almost certainly No. But this is mostly a question about the creative engineering tricks and inventions that are possible given the known laws of physics; it is not a question about physics itself. If we measure the age of objects and materials, we must look at the time-dependent changes of these objects and materials. Chemistry doesn't bring us too far and nuclear physics is the way to go - we're back to various types of radioactive dating.
As you can see, I probably don't understand the question - and I would bet that I will end up being in the majority - so I have no way to predicting whether the text above will be found satisfactory by HDE.