Is the calculated age of the universe that of the visible universe or the entire universe? I dont know how the age is calculated but if it is believed that all we see visibly IS the entire universe then that means it is finite. If it is not finite then the light from the really far astronomcical objects has not reached us yet so there is a possibility it is infinite. Can the method used to calculate the universe age since the Big Bang guarantee that it has an age or is it possible that it existed infinitely?
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It is a common point of confusion, but an infinite universe can still have been at a unique singularity in space-time at a finite time in the past.
This singularity is not a "point in space", since every point in the universe participated in the big-bang. It is merely a time at which the scale factor, which multiplies the separation between all spatial points in the universe, was zero.
So in cosmology governed by the Friedmann equation, the age of the universe thus calculated is the same whether you are talking about the "visible" universe or an infinite universe. The light travel time really doesn't come into this, because different parts of even a very large universe can be causally separated yet still have originated at the same singularity.
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The simplest assumption about the global properties of the Universe is that it looks the same outside the part that is observable to us, as it does inside. That is, we see a finite part of a Universe that is (probably) infinite in extend. If so, then the calculated age — which is finite — applies to all of the Universe, not just the observable part.
The age is calculated on the basis of the observed expansion rate, and the observed densities of the constituents of the Universe. It is possible to imagine a universe with the right mixture of constituents that has existed forever$^{\dagger}$, but for our particular Universe, this just doesn't seem to be the case; it is ruled out by observations.
As a first-order approximation, you can simply take the age $t_\mathrm{Uni}$ to be the reciprocal of the expansion rate $H_0 = 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1} = 2\times10^{-18}$ s. That is,
$$t_\mathrm{Uni} \sim \frac{1}{H_0} = 14\,\mathrm{billion\,years}.$$
However, this assumes that the Universe has been expanding at the same rate throughout its entire history, which is hasn't. More generally, the age is calculated from integrating (numerically except for simplified approximations) the Friedmann equation, yielding 13.819 billion years.
I should say that the calculated age is the time from the Big Bang till now. I guess the safest thing to say is that we don't know what happened the first tiny fraction of a second or so after creation, and in principle it could have existed before this instant, collapsed, and then re-expanded. But no observations I know of suggest this.
$^\dagger$An example of a temporally infinite universe is one containing energy only in the form of a cosmological constant. In this case, the Friedmann equation reduces to $da/dt=aH_0$, with $a$ the scale factor ("size") of the universe, the solution of which is an exponential function with zero size only at $t = -\infty$.
